Let \(\bb{\G}\) be an algebraic group of exceptional Lie type in characteristic \(p\), \(\G=\bb{\G}^{\sigma}\) its fixed-point subgroup under the action of a Steinberg endomorphism \(\sigma\), and \(\overline{\G}\) an almost simple group with socle \(\G\).

The possible classes of maximal subgroups of \(\overline{\G}\) are known, and one of them is made of almost simple subgroups \(M<\overline{\G}\) such that the socle \(H\) of \(M\) is not isomorphic to a group of Lie type in characteristic \(p\); such subgroups are called non-generic.

A finite subgroup \(H\) of \(\bb{\G}\) is Lie primitive if it does not lie in any proper closed positive-dimensional subgroup of \(\bb{\G}\); it is Lie imprimitive if it lies in a positive-dimensional subgroup \(\bb{\X}\) of \(\bb{\G}\); it is strongly imprimitive if \(\bb{\X}\) can be chosen to be stable under the action of \(N_{\aut\bb{\G}}(H)\), where \(\aut\bb{\G}\) is the group generated by inner, diagonal, graph, and field automorphisms of \(\bb{\G}\).

The list of non-generic simple subgroups that may give rise to a maximal subgroup of \(\overline{\G}\) is known, and include Lie primitive subgroups.

We study the possible embeddings of a non-generic primitive simple group \(H\) in the adjoint algebraic group \(\bb{\G}\) in characteristic coprime to \(\size{H}\) when \((\bb{\G},H)\) is one of \((\bb{\F}_4,\PSL_2(25))\), \((\bb{\F}_4,\PSL_2(27))\), \((\bb{\E}_7,\PSL_2(29))\), \((\bb{\E}_7,\PSL_2(37))\). In particular, we construct copies of \(H\) in \(\G\) over a suitable finite field \(k\), and use them to deduce information about the number of conjugacy classes of \(H\) in \(\G\) and \(\bb{\G}\), and about whether \(N_{\overline{\G}}(H)\) is a maximal subgroup of \(\overline{\G}\).

We also study the case of \(H\simeq\alt_6\simeq\PSL_2(9)\) when \(\bb{\G}\) is one of \(\bb{\F}_4\) and \(\bb{\E}_6\) in characteristic coprime to \(\size{H}\), and show that in such cases \(H\) is a strongly imprimitive subgroup of \(\bb{\G}\); in particular, \(N_{\overline{\G}}(H)\) is not a maximal subgroup of \(\overline{\G}\).

One of the main areas of research in group theory is the classification of maximal subgroups of a finite group.

As shown in <cit.>, the maximal subgroups of any finite group can be deduced if we know the maximal subgroups and 1-cohomology of all finite almost simple groups; this thesis is a contribution to the former of these goals.

Going through the list of simple groups, we have that the maximal subgroups of the sporadic simple groups are known except for some cases of the Monster, while for the alternating groups they are classified by the O'Nan-Scott theorem <cit.>. For the classical groups of Lie type, they are classified by Aschbacher's theorem <cit.> (see also <cit.>), in particular they are divided into two categories: the class \(\mathcal{C}\) of maximal subgroups that arise naturally in a geometrical way, stabilising a certain structure of the underlying module, and the class \(\mathcal{S}\) which consists of all the other cases, that is almost simple groups acting irreducibly on the natural module. In particular, for groups of small rank we have a complete list in <cit.>.

The remaining case is the maximal subgroups of exceptional groups of Lie type. Let \(\bb{\G}\) be an algebraic group of exceptional Lie type in characteristic \(p\), \(\G=\bb{\G}^{\sigma}\) its fixed-point subgroup under the action of a Steinberg endomorphism \(\sigma\), and \(\overline{\G}\) an almost simple group with socle \(\G\).

When \(\G=\G(q)\) is not of type \(\F_4\), \(\E_6\), \(\!\prescript{2}{}\E_6\), \(\E_7\), \(\E_8\), the maximal subgroups are classified (see for example <cit.>), so we focus on the remaining cases.

The literature on the topic is explored in <ref>; broadly speaking, all maximal subgroups of \(\overline{\G}\) are known except for a list of almost simple subgroups \(M\). Let \(H\) be the socle of such \(M\), then we have the following classification of \(H\):

If \(H\) is isomorphic to a group of Lie type over a field of characteristic \(p\), then \(H\) is called a generic subgroup; if \(H\) is any other type of finite simple group, it is called a non-generic subgroup.

If \(H\) is contained in a positive-dimensional subgroup \(\bb{\X}\) of \(\bb{\G}\), then \(H\) is called Lie imprimitive. Let \(\aut\bb{\G}\) be the group generated by inner, diagonal, graph, and field automorphisms of \(\bb{\G}\); if \(\bb{\X}\) can be taken to be stable under the action of \(N_{\aut\bb{\G}}(H)\), then \(H\) is called strongly imprimitive in \(\bb{\G}\). If \(H\) is not contained in any positive-dimensional subgroup of \(\bb{\G}\), then it is called Lie primitive.

Generic subgroups are generally well-understood <cit.>, so we focus on the non-generic subgroups. Thanks to the work of several authors, in <cit.> we have a complete (finite) list of non-generic simple groups \(H\) that admit an embedding into an exceptional algebraic group, and for which characteristics such embeddings exist.

Further work by Litterick <cit.> studies the primitivity of the possible embeddings of such groups \(H\) in \(\bb{\G}\) in characteristic \(p\), and divide the triples \((\bb{\G},H,p)\) in three classes:

* Any subgroup \(S\) of \(\bb{\G}\) isomorphic to \(H\) is strongly imprimitive.

* Any subgroup \(S\) of \(\bb{\G}\) isomorphic to \(H\) is primitive.

* A subgroup \(S\) of \(\bb{\G}\) may be primitive, depending on the embedding.

The three lists of cases are contained in <cit.>.

In particular, he shows that if \(H\) is strongly imprimitive in \(\bb{\G}\) then \(N_{\overline{\G}}(H)\) is not a maximal subgroup of \(\overline{\G}\).

Therefore one problem is completing the study of \(H\) in the remaining cases, in particular we would like to know how many conjugacy classes of \(H\) there are in \(\G\) and \(\bb{\G}\), what is \(N_{\overline{G}}(H)\) and whether it is a maximal subgroup of \(\overline{\G}\).

We will see that if \(H\) is Lie primitive, then \(N_{\G}(H)\) is a maximal subgroup of \(\G\) unless it is contained in a subgroup of the same type as \(\G\) (e.g. \(\E_6(p)<\E_6(p^2)\)) or in \(N_{\G}(K)\) for some other Lie primitive subgroup \(K\).

Furthermore, if \(N_{\G}(H)\) is a maximal subgroup of \(\G\) then \(N_{\overline{\G}}(H)\) is a maximal subgroup of \(\overline{\G}\); otherwise, \(N_{\overline{\G}}(H)\) may still be a maximal subgroup of \(\overline{\G}\), in which case it is called a novelty maximal subgroup.

Observe that most of the candidate subgroups \(H\) are “small” groups, like alternating groups or groups of type \(\A_1\). Some of these cases have been studied, for example most of the alternating groups have been dealt with by Craven <cit.>. At the time of writing, his ongoing work completed the study of the remaining cases for \(\F_4\), and almost completed the cases for groups of type \(\E_6\) and \(\E_7\). Other groups of this class have been studied for example in <cit.>, where they study the embeddings of \(\PSL_2(41)\) and \(\PSL_2(49)\) in \(\bb{\E}_8(\mathbb{C})\), and in a similar fashion in other works like <cit.>.

We decided to see whether the approach used in <cit.> could be used also for other cases; we will see that for technical reasons its scope is limited, but nonetheless we could verify some of the existing results. The new results we obtained using this technique can be summarised by the following:

Let \(\bb{\G}\) be an adjoint algebraic group of exceptional Lie type in characteristic \(p\), \(\G=\bb{\G}^{\sigma}\) its fixed-point subgroup under the action of a Steinberg endomorphism \(\sigma\), i.e. \(\G=\G(q)\) for some power \(q\) of \(p\), and \(\overline{\G}\) an almost simple group with socle \(\G\).
Let \((\bb{\G},H)\) be one of \((\bb{\F}_4,\PSL_2(25))\), \((\bb{\F}_4,\PSL_2(27))\), \((\bb{\E}_7,\PSL_2(29))\), \((\bb{\E}_7,\PSL_2(37))\).

If \(\G\), \(N_{\G}(H)\), \(p\), \(q\) appear in <ref>, then \(N_{\G}(H)\) is a maximal subgroup of \(\G\). Furthermore, the number of \(\G\text{-conjugacy}\) classes of \(N_{\G}(H)\) in \(\G\) is given by \(n(\G)\), while Stab denotes which outer automorphisms of \(\G\) stabilise \(N_{\G}(H)\).

Furthermore, the number of \(\bb{\G}\text{-conjugacy}\) classes \(n(\bb{\G})\) is \(n(\G)\) when \(\G\) is \(\F_4\), and \(n(\G)/2\) when \(\G\) is \(\E_7\).

The number of \(\G\text{-}\) and \(\bb{\G}\text{-conjugacy}\) classes of embeddings of \(N_{\G}(H)\) in \(\G\) and \(\bb{\G}\) is \(2\cdot n(\G)\) and \(2\cdot n(\bb{\G})\), respectively.

Since we are considering adjoint groups, when taking the simple group \(\E_7\) \(n(\G)\) must be doubled, as the diagonal morphism fuses classes in pairs.

\(\G\)   \(N_{\G}(H)\)   \(p\)   \(q\)   \(n(\G)\)   Stab

\(\F_4\)   \(\PSL_2(25).2\)   \(p\ne2,3,5\)   \(p\)   1   1

\(\F_4\)   \(\PSL_2(27)\)   [c]\(p\ne2,3,7\)
\(p\equiv\pm1\bmod7\)   \(p\)   1   1

\(\F_4\)   \(\PSL_2(27)\)   [c]\(p\ne2,3,7\)
\(p\equiv\pm2\bmod7\)   \(p^3\)   1   \(\gen{\phi}\)

\(\E_7\)   \(\PSL_2(29)\)   [c]\(p\ne2,3,5,7,29\)
\(p\equiv\pm1\bmod5\)
\(p\equiv\pm1,\pm4,\pm5,\pm6,\pm7,\pm9,\pm13\bmod29\)   \(p\)   4   1

\(\E_7\)   \(\PSL_2(29)\)   [c]\(p\ne2,3,5,7,29\)
\(p\equiv\pm1\bmod5\)
\(p\equiv\pm2,\pm3,\pm8,\pm10,\pm11,\pm12,\pm14\bmod29\)   \(p^2\)   4   \(\gen{\phi}\)

\(\E_7\)   \(\PSL_2(29)\)   [c]\(p\ne2,3,5,7,29\)
\(p\equiv\pm2\bmod5\)   \(p^2\)   4   1

\(\E_7\)   \(\PSL_2(37)\)   [c]\(p\ne2,3,19,37\)
\(p\equiv\pm1,\pm3,\pm4,\pm7,\pm9,\pm10,\pm11,\pm12,\pm16\bmod37\)   \(p\)   2   1

\(\E_7\)   \(\PSL_2(37)\)   [c]\(p\ne2,3,19,37\)
\(p\equiv\pm2,\pm5,\pm6,\pm8,\pm13,\pm14,\pm15,\pm17,\pm18\bmod37\)   \(p^2\)   2   \(\gen{\phi}\)

Conjugacy classes of some primitive subgroups of \(\G(q)\), \(q\) power of \(p\); \(\phi\) is a generator of the field automorphism. The group \(\PSL_2(25).2\) is the extension by the field-diagonal automorphism.

We emphasise that the restriction on \(p\) generally means that the we could not prove anything in other cases, and not, say, that the embedding does not exist or is not primitive.

We also tackled the case of the embedding of \(\alt_6\) in \(\bb{\G}\) when \(\bb{\G}\) is one of \(\bb{\F}_4\) and \(\bb{\E}_6\), for which only partial results are known, see for example <cit.>. Despite \(\alt_6\) being a group of type \(\A_1\) via the isomorphism \(\alt_6\simeq\PSL_2(9)\), the technique we used to obtain the results of <ref> could not be applied. Therefore we used a completely different approach that exploits the geometry of \(\bb{\E}_6\) and the Dickson 3-form on its natural module, and the fact that \(\bb{\F}_4<\bb{\E}_6\). From <cit.> we know that most possible embeddings of \(\alt_6\) in \(\bb{\G}\) are strongly imprimitive, and we know the action of such \(\alt_6\) on the minimal and adjoint module of \(\bb{\G}\). It is known that there is an embedding \(\alt_6<\bb{\X}<\bb{\G}\), where \(\bb{\X}\) is a positive-dimensional subgroup, such that \(\alt_6\) has the required action (see for example <cit.>); we show that any embedding with the required action makes \(\alt_6\) always lie in a proper positive-dimensional subgroup \(\bb{\X}\) of \(\bb{\G}\), making \(\alt_6\) imprimitive. We then prove that \(\alt_6\) is strongly imprimitive:

\(\alt_6\) is strongly imprimitive in \(\bb{\F}_4\) in characteristic \(\ne2,3\), and in \(\bb{\E}_6\) in characteristic \(\ne2,3,5\).

<ref> outline some general knowledge about groups of Lie type, the main objects we work with, and their structure.

<ref> focus on <ref> and explore in more details some topics required to understand how to find a \(\PSL_2(q)\) subgroup of a group of Lie type.

In <ref> we talk about forms over a vector space, with particular focus on the unique symmetric trilinear \(\E_6\text{-invariant}\) form defined over the minimal module of \(\E_6\). These results will be used to prove <ref>.

As previously mentioned, in <ref> we present some of the literature about maximal subgroups of exceptional groups of Lie type, finite and algebraic, in order to contextualise the cases we are studying.

<ref> is dedicated to the proof of <ref>, where the results for each choice of \((\bb{\G},H)\) are contained in a dedicated section; <ref> outlines the general steps we will follow for each of the cases. The algorithms used in the computations are described in <ref>.

<ref> is dedicated to the proof of <ref>, where each choice of \(\bb{\G}\) has its own tailored approach. The algorithms used in the computations are described in <ref>.

The supplementary materials include the files required to reproduce the computations described in <ref>; a complete list is given in <ref>. Since all our computations are performed in , in <ref> we included some information on the Lie algebras and groups constructed using built-in  functions, for those unfamiliar with the software.

Background material

Linear algebraic groups
The contents of this section can be found in <cit.> and <cit.>, unless stated otherwise.

Let \(k\) be a field and consider the affine space \(k^n\).

Let \(S\) be any subset of the polynomial ring \(k[X_1,\ldots,X_n]\). The algebraic set defined by \(S\) is the set
\[\bb{V}(S)\coloneqq\set{(x_1,\ldots,x_n)\in k^n|f(x_1,\ldots,x_n)=0\;\forall f\in S}.\]
Conversely, a subset of \(k^n\) is called algebraic if it is of the form \(\bb{V}(S)\) for some \(S\subseteq k[X_1,\ldots,X_n]\).

Using algebraic sets we can define a topology on \(k^n\).

The open sets of the Zariski topology on \(k^n\) are the subsets \(X\subseteq k^n\) such that \(k^n\backslash X\) is an algebraic set.

In particular, the algebraic sets of \(k^n\) are the closed sets of the topology, and every algebraic set is a topological space with the induced topology.

We can define direct products of algebraic sets in a natural way: let \(V=\bb{V}(S)\), \(S\subseteq k[X_1,\ldots,X_n]\), and \(W=\bb{V}(T)\), \(T\subseteq k[Y_1,\ldots,Y_m]\); if we identify \(k^n\times k^m\) with \(k^{n+m}\), and \(S,T\) with subrings of \(k[X_1,\ldots,X_n,Y_1,\ldots,Y_m]\), then we can define \(V\times W\coloneqq\bb{V}(S\cup T)\), which makes it an algebraic set of \(k^{n+m}\).

We say that a map \(\varphi:V\rightarrow W\) between non-empty algebraic sets \(V,W\subseteq k^n\) is regular if there exist \(f_1,\ldots,f_n\in k[X_1,\ldots,X_n]\) such that, for all \(x\in V\), \(\varphi(x)=(f_1(x),\ldots,f_n(x))\). Observe that such a map is continuous in the Zariski topology.

We can now define algebraic groups, which are the objects we work with.

Take the space \(M_n(k)\) of \(n\times n\) matrices over \(k\), which can be identified with \(k^{n^2}\), and let \(\mu:M_n(k)\times M_n(k)\rightarrow M_n(k)\) be the usual matrix multiplication, which is a regular map.

\(M_n(k)\) equipped with \(\mu\) is called the general linear algebraic monoid of degree \(n\), with identity element the identity matrix \(\mathbb{1}_n\).

A linear algebraic monoid is an algebraic subset \(G\subseteq M_n(k)\) that is closed under \(\mu\) and such that \(\mathbb{1}_n\in G\).

If every element \(A\in G\) has an inverse under \(\mu\) and the map \(\iota:G\rightarrow G\), \(A\mapsto A^{-1}\), is regular, then \(G\) is called a linear algebraic group.

* The multiplicative group \(\bb{\G}_m\coloneqq(k^{\times},\cdot)\) is an algebraic group, as it can be identified with the set
\[\Set{(x,y)\in k^2|xy=1},\]
with component-wise multiplication, and it is an algebraic set defined by \(XY-1\in k[X,Y]\).

* The special linear group \(\SL_n(k)\) is a linear algebraic group, being the algebraic set
\[\bb{V}(\det-1)\subseteq M_n(k),\]
which is closed under multiplication; one can check that the inverse map is regular.

* The general linear group \(\GL_n(k)\) is also a linear algebraic group, as it can be identified with the closed subset
\[\Set{(A,x)\in M_n(k)\times k|\det A\cdot x=1},\]
via the map \(A\mapsto(A,\det A^{-1})\), with component-wise multiplication.

* Any closed subgroup of \(\GL_n(k)\) is itself a linear algebraic group, and the converse is also true: any linear algebraic group can be embedded as a closed subgroup into \(\GL_n(k)\) for some \(n,k\).

An algebraic group is called a torus if it is isomorphic to a direct product of copies of the multiplicative group \(k^{\times}\).

We also introduce the notion of character:

Let \(G\) be a linear algebraic group. A character of \(G\) is a morphism of algebraic groups \(\chi:G\rightarrow\bb{\G}_m\); the set of characters of \(G\) is denoted by \(X(G)\).

Similarly, a cocharacter of \(G\) is a morphism of algebraic groups \(\gamma:\bb{\G}_m\rightarrow G\); the set of cocharacters of \(G\) is denoted by \(Y(G)\).

We have that \(X(G)\) is an abelian group with respect to
\[(\chi_1+\chi_2)(g)\coloneqq\chi_1(g)\chi_2(g)\qquad\chi_1,\chi_2\in X(G),g\in G.\]

Similarly, if \(G\) is abelian then \(Y(G)\) is also an abelian group with respect to

\[(\gamma_1+\gamma_2)(x)\coloneqq\gamma_1(x)\gamma_2(x)\qquad\gamma_1,\gamma_2\in Y(G),x\in\bb{\G}_m.\]

With a topology in place, we can define the following notions:

Let \(U\) be a topological space. Then:

* \(U\) is irreducible if it cannot be decomposed as \(U=U_1\cup U_2\), where \(U_1,U_2\) are non-empty proper closed subsets of \(U\).

* \(U\) is connected if it cannot be decomposed as \(U=U_1\sqcup U_2\) (disjoint union), where \(U_1,U_2\) are non-empty closed subsets of \(U\).

Any irreducible set is also connected, while the converse is in general not true. However, in the case of linear algebraic group the two properties are equivalent (see for example <cit.>).

Let \(V\) be a finite-dimensional vector space over an algebraically closed field \(k\).

An endomorphism \(a\in\Endo V\) is called semisimple if it is diagonalisable, nilpotent if \(a^m=0\) for some \(m\in\mathbb{N}\), and unipotent if \(a-1\) is nilpotent.

Let \(\bb{\G}\) be a linear algebraic group over an algebraically closed field \(k\). Then:

* For any embedding \(\rho:\bb{\G}\rightarrow\GL(V)\) and for any \(g\in\bb{\G}\), there exist \(g_s,g_u\in \bb{\G}\) such that \(g=g_sg_u=g_ug_s\), where \(\rho(g_s)\) is semisimple (diagonalisable), and \(\rho(g_u)\) is unipotent (\(g_u-1\) is nilpotent).

* The above decomposition is independent of the choice of \(\rho\).

See for example <cit.>.

We denote by \(\bb{\G}_u\) (resp. \(\bb{\G}_s\)) the subset of unipotent (resp. semisimple) elements of \(\bb{\G}\).

The radical \(R(\bb{\G})\) of the linear algebraic group \(\bb{\G}\) is the maximal closed connected soluble normal subgroup of \(\bb{\G}\).

\(R(\bb{\G})_u\), or \(R_u(\bb{\G})\), the unipotent radical of \(\bb{\G}\), is the maximal closed connected normal unipotent subgroup of \(\bb{\G}\).

\(\bb{\G}\) is called reductive if \(R_u(\bb{\G})=1\), and semisimple if it is connected and \(R(\bb{\G})=1\).

A non-trivial semisimple algebraic group \(\bb{\G}\) is called simple if it has no non-trivial proper closed connected normal subgroup.

BN-pairs and Steinberg endomorphisms
The contents of this section can be found in <cit.> and <cit.>, unless specified otherwise.

The groups with a BN-pair are an important class, which also includes the groups of Lie type we are interested in.

An abstract group \(G\) is said to possess a BN-pair if there are \(B,N\le G\) such that:

* \(\gen{B,N}=G\).

* \(H\coloneqq B\cap N\unlhd N\), and \(W\coloneqq N/H\) is a finite group generated by a set \(S\) of involutions.

* \(n_sBn_s\ne B\), where \(n_s\in N\) maps to \(s\in S\) under the natural homomorphism \(N\rightarrow W\).

* \(n_sBn\subseteq Bn_snB\cup BnB\) for all \(s\in S\) and \(n\in N\).

* \(\displaystyle\bigcap_{n\in N}nBn^{-1}=H\).

\(W\) is called the Weyl group of \(G\).

We have not only that \(G=BNB\), but also the following holds:

For any \(w\in W\), let \(n_w\) be a representative of \(w\) in \(N\). Then we have a double-coset decomposition

\[G=\bigsqcup_{w\in W}Bn_wB.\]

A consequence of this result is the following:

\(B\) is self-normalising in \(G\).

A BN-pair as in the above definition is called split if there exists \(U\unlhd B\) such that:

* \(B=U\rtimes H\).

* For any \(n\in N\), \(U^n\cap B\subseteq U\).

For the rest of the section, let \(\bb{\G}\) be an algebraic group over an algebraically closed field \(k\).

\(\bb{\G}\) has a reductive BN-pair if it has a split BN-pair such that:

* \(H\) is a torus and it is self-centralising in \(\bb{\G}\).

* \(U\) is closed connected and nilpotent.

Note that the exceptional groups of Lie type, which we are interested in, have a reductive BN-pair.

An important class of subgroups is the following:

A subgroup \(B\) of \(\bb{\G}\) is called a Borel subgroup if it is a maximal closed connected soluble subgroup.

The following result describes an important relation between Borel subgroups and tori:

Let \(\bb{\G}\) be a connected reductive algebraic group, \(B\le\bb{\G}\) a Borel subgroup, \(\bb{\T}\le B\) a maximal torus. Let \(N\coloneqq N_{\bb{\G}}(\bb{\T})\), then \(B, N\) is a BN-pair in \(\bb{\G}\).

Observe that by <ref> this implies that if \(\bb{\T}\) is a maximal torus of \(\bb{\G}\) and \(N=N_{\bb{\G}}(\bb{\T})\), then \(N=\bb{\T}.W\), where \(W\) is the Weyl group of \(\bb{\G}\).

Let \(\bb{\G}\) be a connected reductive algebraic group, \(B\) a Borel subgroup of \(\bb{\G}\), and \(S\) a set of involutions generating the Weyl group \(W\) of \(\bb{\G}\).

A standard parabolic subgroup of \(\bb{\G}\) is a subgroup \(P_I\coloneqq BN_IB\), where \(I\subseteq S\), \(W_I\coloneqq\gen{s|s\in I}\le W\), and \(N_I\) is a set of representatives in \(N\) of \(W_I\) under the natural homomorphism \(N\rightarrow W\).

A parabolic subgroup of \(\bb{\G}\) is any conjugate of a standard parabolic subgroup.

Under the same conditions, we have the following results:

* For any \(I\subseteq S\), \(P_I\) is a closed connected self-normalising subgroup of \(\bb{\G}\) that contains \(B\).

* The subgroups \(P_I\) are mutually non-conjugate, and for any \(I,J\subseteq S\) we have \(P_I\cap P_J=P_{I\cap J}\).

* Any overgroup of \(B\) in \(\bb{\G}\) arises in this way.

See for example <cit.>.

* The subgroup \(B\) of \(\bb{\G}\) defining a BN-pair is a Borel subgroup of \(\bb{\G}\).

* All Borel subgroups of \(\bb{\G}\) are conjugate.

In particular, we have that a subgroup of \(\bb{\G}\) is parabolic if and only if it contains a Borel subgroup.

See for example <cit.> and <cit.>.

This implies a similar result for tori.

Let \(\bb{\G}\) be a linear algebraic group. A torus \(\bb{\T}\) of \(\bb{\G}\) is a maximal torus if it is maximal among the tori of \(\bb{\G}\) with respect to inclusion.

Let \(\bb{\G}\) be a linear algebraic group, then all maximal tori of \(\bb{\G}\) are conjugate.

See <cit.>; this follows from the fact that they are connected soluble subgroups hence contained in some Borel subgroup, which are all conjugate by <ref>.

The rank of \(\bb{\G}\), denoted by \(\rk\bb{\G}\), is the dimension of a maximal torus \(\bb{\T}\) of \(\bb{\G}\), that is \(\bb{\T}\simeq {(k^{\times})}^{\rk\bb{\G}}\).

Now let \(\phi:G\rightarrow G\) be an isomorphism of abstract groups such that:

* \(\phi(U)=U\), \(\phi(H)=H\), \(\phi(N)=N\).

* If \(\phi(Hn)\subseteq Hn\) then \(Hn\) contains a fixed point of \(\phi\).

Then the fixed-point set \(G^{\phi}\) has a split BN-pair given by \(B^{\phi}\) and \(N^{\phi}\), in particular we have that \(B^{\phi}=U^{\phi}\rtimes H^{\phi}\) and \(H^{\phi}=B^{\phi}\cap N^{\phi}\), and the Weyl group of \(G^{\phi}\) is \(W^{\overline{\phi}}\), where \(\overline{\phi}:W\rightarrow W\) is the homomorphism induced from \(\phi\) using the fact that \(N\) and \(H\) are invariant under \(\phi\).

We are interested about this behaviour when \(\phi\) belongs to the following class of morphisms:

Let \(k=\overline{\mathbb{F}}_q\), and \(q=p^r\). The standard Frobenius map with respect to \(\mathbb{F}_q\) is defined as

\[F_q\colon k^n\rightarrow k^n,\qquad (x_1,\ldots,x_n)\mapsto(x_1^q,\ldots,x_n^q).\]

This notion can be extended to algebraic groups defined over \(\overline{k}\), via the corresponding affine variety.

A homomorphism \(\sigma:\bb{\G}\rightarrow\bb{\G}\) of algebraic groups is called a Steinberg endomorphism if some power of \(\sigma\) is a Frobenius map.

Let \(\bb{\G}=\GL_n(k)\) for \(k=\overline{F}_q\), \(q=p^r\). A reductive BN-pair of \(\bb{\G}\) can be obtained by taking for \(B\) the upper triangular matrices, and for \(N\) the monomial matrices. Then, \(H\) is made of diagonal matrices, \(U\) of upper unitriangular matrices, while \(W\) can be identified with of monomial matrices with entries only 0 or 1, hence it is isomorphic to \(\sym(n)\).

The standard Frobenius endomorphism of \(\bb{\G}\) with respect to \(\mathbb{F}_q\) is the map
\[F_q\colon\bb{\G}\rightarrow\bb{\G},\qquad(a_{ij})\mapsto(a_{ij}^q),\]

induced by letting the standard Frobenius map act on the matrix entries of an element of \(\bb{\G}\).

The fixed point space of \(F_q\) is the finite general linear group \(\bb{\G}^{F_q}=\GL_n(q)\), for which \(B^{F_q}\) and \(N^{F_q}\) form a BN-pair.

We have the following dichotomy for endomorphisms of simple algebraic groups:

Let \(\bb{\G}\) be a simple linear algebraic group, \(\sigma:\bb{\G}\rightarrow\bb{\G}\) an endomorphism. Then exactly one of the following holds:

* \(\sigma\) is an automorphism of algebraic groups.

* The group of fixed points \({\bb{\G}}^{\sigma}\) is finite.

The second case happens if and only if \(\sigma\) is a Steinberg endomorphism.

In regard to Steinberg endomorphisms, we also mention the Lang-Steinberg theorem, which is often used to transfer statements about an algebraic group \(\bb{\G}\) to finite groups of the form \(\bb{\G}^{\sigma}\) for a Steinberg endomorphism \(\sigma\) of \(\bb{\G}\).

Let \(\bb{\G}\) be a connected linear algebraic group over \(k=\overline{\mathbb{F}}_p\) with a Steinberg endomorphism \(\sigma:\bb{\G}\rightarrow\bb{\G}\). Then the morphism \(F\colon\bb{\G}\rightarrow\bb{\G}\) defined by \(g\mapsto\sigma(g)g^{-1}\) is surjective.

Root systems

The results in this section can be found in <cit.>, unless stated otherwise.

Let \(E\) be a finite-dimensional real vector space. A subset \(\Phi\subset E\) is called a root system if:

* \(\Phi\) is finite, \(0\not\in\Phi\), \(\gen{\Phi}=E\).

* If \(\alpha,c\alpha\in\Phi\), then \(c=\pm1\).

* For any \(\alpha\in\Phi\) there is a reflection \(w_{\alpha}\in\GL(E)\) along the hyperplane orthogonal to \(\alpha\) that stabilises \(\Phi\).

* For any \(\alpha,\beta\in\Phi\), \(w_{\alpha}(\beta)-\beta\) is an integral multiple of \(\alpha\).

The group \(W=W(\Phi)\coloneqq\gen{w_{\alpha}\mid\alpha\in\Phi}\) is the Weyl group of \(\Phi\), while \(\dim E\) is the rank of \(\Phi\).

Let \(\Phi\) be a root system. A subset \(\Delta\subset\Phi\) is called a base, and its elements are called fundamental roots, if it is a basis for \(E\) and any root \(\beta\in\Phi\) can be written as \(\beta=\sum_{\alpha\in\Delta}c_{\alpha}\alpha\) such that the \(c_{\alpha}\) are either all non-negative or all non-positive. The roots with non-negative coefficients form a positive system \(\Phi^+\subset\Phi\), and \(\Phi=\Phi^+\sqcup\Phi^-\), where \(\Phi^-\coloneqq-\Phi^+\) is the set of roots with non-positive coefficients. We also define the height of a root \(\beta\) as \(\height(\beta)=\sum_{\alpha\in\Delta}c_{\alpha}\).

A subset \(\Phi'\subset\Phi\) is a closed subroot system if it is a root system that is closed under addition with respect to \(\Phi\), i.e. for all \(\alpha,\beta\in\Phi'\), if \(\alpha+\beta\in\Phi\) then \(\alpha+\beta\in\Phi'\).

We can introduce an order relation between the roots in the following way. Let \(E^+\) be the subset of \(E\) satisfying the following conditions:

* If \(v\in E^+\) and \(\lambda>0\), then \(\lambda v\in E^+\).

* If \(v_1,v_2\in E^+\), then \(v_1+v_2\in E^+\).

* For all \(v\in E\), exactly one of \(v\in E^+\), \(-v\in E^+\), \(v=0\), holds.

We then say that \(v_2\prec v_1\) when \(v_1-v_2\in E^+\), which induces a total ordering on \(E\).

In particular, we have that a positive system of roots \(\Phi^+\) of \(\Phi\) has the form \(\Phi^+=\Phi\cap E^+\) for some total ordering of \(E\).

With the ordering on \(\Phi\) induced from \(E\), there exists a unique \(\alpha_0\in\Phi\) such that \(\alpha\prec\alpha_0\) for all \(\alpha_0\ne\alpha\in\Phi\).

See for example <cit.>.

The root \(\alpha_0\in\Phi\) described by <ref> is the highest root of \(\Phi\).

Note that bases and positive root systems are in bijection, and any two bases (or positive systems) of a root system are conjugate under its Weyl group.

Since \(\Phi\) is finite, generates \(E\), and is stabilised by \(W\), then \(W\) is always finite, and stabilises a positive-definite \(W\text{-invariant}\) symmetric bilinear form \((-,-)\) on \(E\), and we may assume it makes \(E\) Euclidean <cit.>.

A non-empty root system \(\Phi\) with base \(\Delta\) is decomposable if there exists a non-trivial partition of \(\Delta\) into two subsets orthogonal with respect to \((-,-)\). Otherwise, it is indecomposable.

We can define the dual root system \(\Phi^{\vee}=\Set{\alpha^{\vee}|\alpha\in\Phi}\), where the coroot \(\alpha^{\vee}\) is defined by \(\alpha^{\vee}\coloneqq 2\alpha/(\alpha,\alpha)\), and write explicitly the action of \(W\) on \(E\): for \(v\in E\), \(w_{\alpha}\in W\) we have

\begin{equation}\label{reflection}
\end{equation}

An important role is played by the map \(n:\Phi\times\Phi\rightarrow\mathbb{Z}\) defined by \(n(\alpha,\beta)\coloneqq 2(\alpha,\beta)/(\beta,\beta)\). In particular, we have \(n(\alpha,\beta)\cdot n(\beta,\alpha)=4\cos^2\angle(\alpha,\beta)\), where \(\angle(\alpha,\beta)\) denotes the angle between \(\alpha,\beta\in E\) with the metric induced by \((-,-)\). This means that for \(\beta\ne\pm\alpha\) the product can be at most 3. We can summarise the relation between the two roots as follows:

\(n(\alpha,\beta)\)     \(n(\beta,\alpha)\)     \(\angle(\alpha,\beta)\)     \(\lVert\beta\rVert^2/\lVert\alpha\rVert^2\)
0     0     \(\pi/2\)     any

1     1     \(\pi/3\)     1

\(-1\)     \(-1\)     \(2\pi/3\)     1

1     2     \(\pi/4\)     2

\(-1\)     \(-2\)     \(3\pi/4\)     2

1     3     \(\pi/6\)     3

\(-1\)     \(-3\)     \(5\pi/6\)     3

In particular, we can classify the 2-dimensional root systems, since the two roots of the base cannot form an acute angle.

We can also deduce the following:

Let \(\Phi\) be an indecomposable root system. Then:

* \(\Phi\) can have at most two different root lengths.

* All roots of \(\Phi\) of the same length are conjugate under the Weyl group \(W(\Phi)\).

See for example <cit.>.

We can define the Cartan matrix and Dynkin diagram associated to a root system \(\Phi\) in the following way. Take a base \(\Delta=\Set{\alpha_1,\ldots,\alpha_l}\) of \(\Phi\), and define \(n_{ij}\coloneqq n(\alpha_i,\alpha_j)\). The Cartan matrix is defined as the \(l\times l\) matrix with entries \((n_{ij})_{i,j}\), while the Dynkin diagram is an undirected graph with \(\alpha_1,\ldots,\alpha_l\) as vertices, and two distinct vertices \(\alpha_i\), \(\alpha_j\) are joined by exactly \(n_{ij}n_{ji}\) edges.

Note that a Dynkin diagram is connected if and only if the corresponding root system is indecomposable, and it is simply laced, i.e. no pair of adjacent vertices is connected by multiple edges, if and only if all roots have the same length.

The only possible connected Dynkin diagrams are the following:
l @ l
(-1,0) node[anchor=east]  \(\A_n\);
in 0,...,5
[xshift=cm,thick] (cm,0) circle (.3cm);
[dotted,thick] (0.3 cm,0) – +(1.4 cm,0);
in 1.15,...,4.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);

(-1,0) node[anchor=east]  \(\B_n\);
in 0,...,4
[xshift=cm,thick,fill=black] (cm,0) circle (.3cm);
[xshift=5 cm,thick] (5 cm, 0) circle (.3 cm);
[dotted,thick] (0.3 cm,0) – +(1.4 cm,0);
in 1.15,...,3.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);
[thick] (8.3 cm, .1 cm) – +(1.4 cm,0);
[thick] (8.3 cm, -.1 cm) – +(1.4 cm,0);

(-1,0) node[anchor=east]  \(\C_n\);
in 0,...,4
[xshift=cm,thick] (cm,0) circle (.3cm);
[xshift=5 cm,thick,fill=black] (5 cm, 0) circle (.3 cm);
[dotted,thick] (0.3 cm,0) – +(1.4 cm,0);
in 1.15,...,3.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);
[thick] (8.3 cm, .1 cm) – +(1.4 cm,0);
[thick] (8.3 cm, -.1 cm) – +(1.4 cm,0);

(-1,0) node[anchor=east]  \(D_n\);
in 0,...,4
[xshift=cm,thick] (cm,0) circle (.3cm);
[xshift=8 cm,thick] (30: 17 mm) circle (.3cm);
[xshift=8 cm,thick] (-30: 17 mm) circle (.3cm);
[dotted,thick] (0.3 cm,0) – +(1.4 cm,0);
in 1.15,...,3.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);
[xshift=8 cm,thick] (30: 3 mm) – (30: 14 mm);
[xshift=8 cm,thick] (-30: 3 mm) – (-30: 14 mm);

(-1,1) node[anchor=east]  \(\E_6\);
in 0,...,4
[thick,xshift=cm] (cm,0) circle (3 mm);
in 0,...,3
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick] (4 cm,2 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 1.4 cm);

(-1,1) node[anchor=east]  \(\E_7\);
in 0,...,5
[thick,xshift=cm] (cm,0) circle (3 mm);
in 0,...,4
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick] (4 cm,2 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 1.4 cm);

(-1,1) node[anchor=east]  \(\E_8\);
in 0,...,6
[thick,xshift=cm] (cm,0) circle (3 mm);
in 0,...,5
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick] (4 cm,2 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 1.4 cm);

(-3,0) node[anchor=east]  \(\F_4\);
[thick,fill=black] (-2 cm ,0) circle (.3 cm);
[thick,fill=black] (0 ,0) circle (.3 cm);
[thick] (2 cm,0) circle (.3 cm);
[thick] (4 cm,0) circle (.3 cm);
[thick] (15: 3mm) – +(1.5 cm, 0);
[xshift=-2 cm,thick] (0: 3 mm) – +(1.4 cm, 0);
[thick] (-15: 3 mm) – +(1.5 cm, 0);
[xshift=2 cm,thick] (0: 3 mm) – +(1.4 cm, 0);

(-1,0) node[anchor=east]  \(\G_2\);
[thick,fill=black] (0 ,0) circle (.3 cm);
[thick] (2 cm,0) circle (.3 cm);
[thick] (30: 3mm) – +(1.5 cm, 0);
[thick] (0: 3 mm) – +(1.4 cm, 0);
[thick] (-30: 3 mm) – +(1.5 cm, 0);

where the filled nodes denote the long roots.

The root system associated to a Dynkin diagram
The Dynkin diagram gives complete information about the root system, by giving the geometric relations between a set of fundamental roots.

In particular, it is possible to obtain all the roots using the following algorithm.

If the Dynkin diagram is simply laced:

* Choose a node and give it a label of 1, give every other node a label of 0; this corresponds to choosing a fundamental root.

*  Find any node such that its label \(l\) is \(<1/2\) the sum \(s\) of the labels of its neighbours, then replace the label with \(s-l\). This new labelling corresponds to a root of the system.

*  Repeat the previous step until no label can be incremented; for Dynkin diagrams, the process terminates. This corresponds to the longest root of the system.

* Return to the first step and repeat the process until all the possible combinations of labellings have been explored.

This gives all the positive roots, so by taking the same labellings with the opposite sign we obtain all the negative roots.

If the Dynkin diagram is not simply laced, step <ref> needs to be tweaked, and neighbouring roots that are longer than the selected root have a higher weight when computing \(s\) (they count twice for doubly linked diagrams, that is \(\B_n, \C_n, \F_4\), and thrice for triply linked diagrams, that is \(\G_2\)). In particular, at step <ref> one obtains the highest root or the highest short root depending on whether the algorithm started with a long or short root, respectively.

We give an example of this algorithm an compute the highest root of \(\F_4\). Consider the Dynkin diagram of \(\F_4\) depicted in <ref>, an label the nodes from left to right as \(\alpha_1,\ldots,\alpha_4\).

Since there are two root lengths, we start wit a long root, say \(\alpha_1\), and label its node with 1, and the other nodes with 0. The following table show consecutive iterations of step (ii) of the algorithm; each line shows the label \(l\) of each node followed by the weighted sum \(s\) of it neighbours, and we highlight a node for which \(l<s/2\) holds.

Observe that since \(\alpha_3\) is a short root and \(\alpha_2\) is a long root, the sum \(s\) for \(\alpha_3\) is the label of \(\alpha_4\) plus twice the label of \(\alpha_2\).

\(\alpha_1\)     \(\alpha_2\)     \(\alpha_3\)     \(\alpha_4\)
1 (0)     0 (1)     0 (0)     0 (0)

1 (1)     1 (1)     0 (2)     0 (0)

1 (1)     1 (3)     2 (2)     0 (2)

1 (1)     1 (3)     2 (4)     2 (2)

1 (2)     2 (3)     2 (6)     2 (2)

1 (2)     2 (5)     4 (6)     2 (4)

1 (3)     3 (5)     4 (8)     2 (4)

2 (3)     3 (6)     4 (8)     2 (4)

Since \(l\ge s/2\) for each node, the process stops, and we find the highest root of \(\F_4\): \(\alpha_0=2\alpha_1+3\alpha_2+4\alpha_3+2\alpha_4\).

All the positive roots of \(\A_n\) are those of the form \(\sum_{i=0}^j\alpha_{k+i}\) where \(\alpha_1,\ldots,\alpha_n\) are a set of fundamental roots, \(1\le k\le n\), and \(0\le j\le n-k\).

This follows directly from the algorithm described above. We start by labelling the node \(k\) with 1, and every other node with 0, thus selecting the root \(\alpha_k\). Now the sum of the neighbouring labels is 0 for every node except for \(k-1\) and \(k+1\) (if \(k>1\) and \(k<n\), respectively), for which the sum is 1. Since they have a label of \(0<1/2\), we can give either of them a label of 1. This corresponds to the roots \(\alpha_{k-1}+\alpha_k\), and \(\alpha_k+\alpha_{k+1}\).

Now assume that after repeating step (ii) in the algorithm described above \(j\) times, \(1\le j<n-1\), we have a root that is of the required form, i.e. \(\sum_{i=0}^j\alpha_{k+i}\), \(1\le k\le n-j\). Then the labels \(l\) and the sums of neighbours \(s\) are as follows:

* For \(i=0,\ldots,k-2\) and \(i=k+j+2,\ldots,n\), \(l=0\) and \(s=0\), assuming \(k\ge2\) and \(k\le n-k-2\), respectively.

* For \(i=k-1\) and \(i=k+j+1\), \(l=0\) and \(s=1\), assuming \(k\ge1\) and \(k\le n-j-1\), respectively.

* For \(i=k\) and \(i=k+j\), \(l=1\) and \(s=1\).

* For \(i=k+1,\ldots,k+j-1\), \(l=1\) and \(s=2\), assuming \(j>1\).

The only cases where \(l<s/2\) are \(i=k-1\) and \(i=k+j+1\), if they are defined, which leads to the roots \(\sum_{i=0}^{j+1}\alpha_{k-1+i}\) and \(\sum_{i=0}^{j+1}\alpha_{k+i}\) respectively, showing that they also have the required form.

When step (ii) is repeated \(n-1\) times, we have the root \(\alpha_1+\ldots+\alpha_n\), and all the labels on the graph are 1. The sum of neighbours for each node is either 1 or 2, thus \(l\ge s/2\) for all nodes, and we cannot proceed any further to find new roots.

As this argument is independent on which starting node \(k\) we choose, we have that all the roots must have the required form.

Let \(\alpha_1,\ldots,\alpha_4\) be a base of \(\F_4\) as in the Dynkin diagram in <ref>, and let \(\alpha_0\) be the highest root of \(\F_4\) described in <ref>.

The maximal closed subroot systems of \(\F_4\) have type \(\A_1\C_3\), \(\A_2\A_2\), and \(\B_4\). Furthermore, up to conjugacy under the action of the Weyl group, they have base \(\left\{-\alpha_0,\alpha_2,\alpha_3,\alpha_4\right\}\), \(\left\{-\alpha_0,\alpha_1,\alpha_3,\alpha_4\right\}\), and \(\left\{-\alpha_0,\alpha_1,\alpha_2,\alpha_3\right\}\), respectively.

See for example <cit.>.

In particular, we can immediately deduce the following:

Let \(\Phi_i=\Phi_i^+\cup-\Phi_i^+\) for \(i=1,2\), where \(\Phi_1^+=\left\{-\alpha_0,\alpha_1,\alpha_1-\alpha_0\right\}\) and \(\Phi_2^+=\left\{\alpha_3,\alpha_4,\alpha_3+\alpha_4\right\}\). Then each \(\Phi_i\) is a subroot system of \(\F_4\) of type \(\A_2\), and \(\Phi_1\cup\Phi_2\) is a subroot system of \(\F_4\) of type \(\A_2\A_2\).

Lie algebras and Chevalley groups
Unless stated otherwise, the contents of <ref> can be found in <cit.>, <ref> can be found in <cit.>.

Lie algebras

A Lie algebra is a vector space \(\mathcal{L}\) on which a product operation \([\cdot,\cdot]\) is defined by the following properties:

* \([\cdot,\cdot]\) is bilinear.

* \([x,x]=0\) for all \(x\in\mathcal{L}\).

* \([[x,y],z]+[[y,z],x]+[[z,x],y]=0\) for all \(x,y,z\in\mathcal{L}\).

It can be easily checked that Lie multiplication is anticommutative. (iii) is known as the Jacobi identity. We sometimes refer to \([x,y]\) as the commutator of \(x\) and \(y\).

If \(\mathcal{U},\mathcal{V}\) are subspaces of \(\mathcal{L}\), let \([\mathcal{U},\mathcal{V}]\) be the subspace of \(\mathcal{L}\) defined by \(\gen{[u,v]\colon u\in\mathcal{U},v\in\mathcal{V}}\). A subspace \(\mathcal{M}\) of \(\mathcal{L}\) is called:

* A subalgebra if \([\mathcal{M},\mathcal{M}]\subseteq\mathcal{M}\).

* An ideal if \([\mathcal{M},\mathcal{L}]\subseteq\mathcal{M}\).

Note that anticommutativity implies that multiplication of subspaces is commutative, therefore every ideal is two-sided. A Lie algebra is called simple if it does not have any non-trivial proper ideal.

A linear map \(\delta:\mathcal{L}\rightarrow\mathcal{L}\) is called a derivation of \(\mathcal{L}\) if \(\delta[x,y]=[\delta x,y]+[x,\delta y]\) for all \(z,y\in\mathcal{L}\).

The set of derivations of \(\mathcal{L}\) is denoted by \(\Der\mathcal{L}\).

We can define the adjoint representation map \(\ad\colon\mathcal{L}\rightarrow\Der\mathcal{L}\) by \(\ad x.y=[x,y]\) for \(y\in\mathcal{L}\); we can verify that \(\ad\) is a derivation using the Jacobi identity:

\begin{align*}
\ad x.[y,z]&=[x,[y,z]]=[[x,y],z]+[[z,x],y]=\\
&=[[x,y],z]+[y,[x,z]]=[\ad x.y,z]+[y,\ad x.z].
\end{align*}

We can use the map \(\ad\) to define a symmetric bilinear form on \(\mathcal{L}\), called Killing form, by \((x,y)=\tr(\ad x.\ad y)\).

A subalgebra \(\mathcal{H}\) of \(\mathcal{L}\) is called a Cartan subalgebra if it satisfies the following:

* \(\overbrace{[[[\m{H},\m{H}],\m{H}]\ldots]}^{r\text{ commutators}}=0\) for some \(r\), that is \(\m{H}\) is nilpotent.

* If \([x,h]\in\m{H}\) for all \(h\in\m{H}\), then \(x\in\m{H}\), i.e. there is no subalgebra \(\m{H}\subset\m{M}\subset\m{L}\) that contains \(\m{H}\) as an ideal.

Any two Cartan subalgebras of a given Lie algebra \(\m{L}\) are isomorphic via some automorphism of \(\m{L}\). Their dimension is called the rank of \(\m{L}\). If \(\m{L}\) is simple, then \([\m{H},\m{H}]=0\).

Cartan decomposition
It can be proved that if a Lie algebra \(\m{L}\) is simple over \(\mathbb{C}\) then there exists a decomposition of \(\m{L}\), called a Cartan decomposition, that has the form

\[\m{L}=\m{H}\oplus\m{L}_{r_1}\oplus\cdots\oplus\m{L}_{r_k},\]

where \(\m{H}\) is a Cartan subalgebra of \(\m{L}\), and for each \(i\) we have \(\dim\m{L}_{r_i}=1\) and \([\m{H},\m{L}_{r_i}]=\m{L}_{r_i}\).

Take now \(0\ne e_r\in\m{L}_r\) and \(h\in\m{H}\), then we can write

\[[h,e_r]=r(h)e_r,\]

where \(r:\m{H}\rightarrow k\). Since \(r\) is linear, then \(r\in\mathcal{H}^*\coloneqq\hom(\m{H},k)\), and the maps \(r_1,\ldots,r_k\) are called roots, since they indeed form a root system, as described in <ref>. The subspaces \(\m{L}_{r_i}\) are then called root spaces of \(\m{L}\) relative to \(\m{H}\).

It can be shown that the Killing form of \(\m{L}\) is non-singular when restricted to \(\m{H}\), therefore for any \(r\in\m{H}^*\) there is a unique \(x\in\m{H}\) such that \(r\) can be written as \(r(h)=(x,h)\), and we may identify the root \(r\colon h\mapsto r(h)\) with the (unique) associated element \(x\) of \(\m{H}\).

Thus, the Cartan decomposition can be written as

\[\m{L}=\m{H}\oplus\bigoplus_{r\in\Phi}\m{L}_r,\]

where \(\Phi\) is the root system associated to \(\m{L}\). Then, for any \(r,s\in\Phi\) we have:

* \([\m{L}_r,\m{L}_s]=\m{L}_{r+s}\), if \(r+s\in\Phi\).

* \([\m{L}_r,\m{L}_s]=0\), if \(0\ne r+s\not\in\Phi\).

* \([\m{L}_r,\m{L}_{-r}]=kr\subseteq\m{H}\).

* \([\m{H},\m{L}_r]=\m{L}_r\).

Note that in (iii) we are identifying \(r\) with the corresponding element \(x\) of \(\m{H}\). It is common practice to not take \(r\) as the generating vector, but its scalar multiple
\[h_r\coloneqq\frac{2r}{(r,r)}.\]

Structure constants
Given a Cartan decomposition of \(\m{L}\), using the above notation, let \(\Pi\subset\Phi\) be a set of fundamental roots, and for each \(r\in\Phi^+\) fix \(0\ne e_r\in\m{L}_r\), then there exists a unique \(e_{-r}\in\m{L}_{-r}\) such that \([e_r,e_{-r}]=h_r\).

The set

\[\left\{e_r,h_s\colon r\in\Phi,s\in\Pi\right\}\]

is a basis for \(\m{L}\), called a Chevalley basis. Its elements multiply as follows:

* \([h_r,h_s]=0\) for \(r,s\in\Pi\), since \([\m{H},\m{H}]=0\).

* \([e_r,e_{-r}]=h_r\) for \(r\in\Phi\), by construction.

* \([e_r,e_s]=0\) for \(r,s\in\Phi\), \(0\ne r+s\not\in\Phi\).

* \([h_r,e_s]=A_{r,s}e_s\) for \(r\in\Pi,s\in\Phi\).

* \([e_r,e_s]=N_{r,s}e_{r+s}\) for \(r,s\in\Phi,r+s\in\Phi\).

\(A_{r,s}\) and \(N_{r,s}\) are field elements that will be defined below.

Let \(\Phi\) be a finite root system, and \(r,s\in\Phi\) with \(s\ne\pm r\). Then there exist \(p,q\ge 0\) such that \(ir+s\in\Phi\) for \(-p\le i\le q\), but \(s-(p+1)r,s+(q+1)r\not\in\Phi\). The sequence of roots
\[s-pr,\ldots,s,\ldots,s+qr\]
is called the \(r\text{-chain}\) of roots through \(s\).

Going back to <ref>, we have that \(A_{r,s}=n(s,r)\) with \(n\) defined in <ref>, so if \(s\ne\pm r\) then (iv) is equivalent to

\[[h_r,e_s]=\frac{2(r,s)}{(r,r)}e_s=(p-q)e_s,\]

where \(p,q\) are integers defining an \(r\text{-chain}\) of roots through \(s\).

In particular, the \(A_{r,s}\) for \(r,s\in\Pi\) are the entries of the Cartan matrix of the root system.

Let \(\Phi\) be a finite root system, \(r,s\in\Phi\), then \(A_{r,-s}=-A_{r,s}\).

This follows from \(A_{r,s}=n(s,r)\) and the definition of \(n(s,r)\), since \((-,-)\) is bilinear.

For \(s\ne\pm r\), one could also deduce it from \(A_{r,s}=p-q\), with \(p,q\) defining a \(r\text{-chain}\) through \(s\). Indeed, this means that \(s-pr,\ldots,s+qr\in\Phi\) and \(s-(p+1)r,s+(q+1)r\notin\Phi\), so by taking the corresponding negative roots we also have \(-s-qr,\ldots,-s+pr\in\Phi\) and \(-s-(q+1)r,-s+(p+1)r\notin\Phi\). Therefore, \(q,p\) define an \(r\text{-chain}\) through \(-s\), so \(A_{r,-s}=q-p=-A_{r,s}\).

The \(N_{r,s}\) are called the structure constants of \(\m{L}\), and depend on the choice of the \(e_r\). They satisfy the following properties:

* \(N_{s,r}=-N_{r,s}\), for \(r,s\in\Phi\).

* \(\frac{N_{r_1,r_2}}{(r_3,r_3)}=\frac{N_{r_2,r_3}}{(r_1,r_1)}=\frac{N_{r_3,r_1}}{(r_2,r_2)}\), for all \(r_1,r_2,r_3\in\Phi\) such that \(r_1+r_2+r_3=0\).

* \(N_{r,s}N_{-r,-s}=-(p+1)^2\), for \(r,s\in\Phi\).

* \(\frac{N_{r_1,r_2}N_{r_3,r_4}}{(r_1+r_2,r_1+r_2)}+\frac{N_{r_2,r_3}N_{r_1,r_4}}{(r_2+r_3,r_2+r_3)}+\frac{N_{r_3,r_1}N_{r_2,r_4}}{(r_3+r_1,r_3+r_1)}=0\), for all \(r_1,r_2,r_3,r_4\in\Phi\) such that \(r_1+r_2+r_3+r_4=0\) and no pair of them are opposite.

Note that in (iii), \(p\) is as described in <ref>.

There is a choice of the \(e_r\) such that \(N_{r,s}=\pm(p+1)\): the sign can be chosen only for a set of ordered pairs \((r,s)\), called extraspecial pairs, and it is then uniquely determined for all the other pairs of roots. Note that such a choice would imply that \(N_{-r,-s}=\mp(p+1)\), by (iii).

An ordered pair \((r,s)\) of roots is a special pair if \(r+s\in\Phi\) and \(0\prec r\prec s\); it is an extraspecial pair if it is a special pair and for any other special pair \((r_1,s_1)\) such that \(r_1+s_1=r+s\) we have \(r\preceq r_1\).

Here, \(\prec\) denotes the order relation introduced in <ref>.

Chevalley groups

Observe that the map \(\ad e_r\) is nilpotent, since using the relations shown in <ref> we have:

\begin{gather*}
\ad e_r.\m{L}_r=0,\\
(\ad e_r)^2.\m{H}=\ad e_r.\m{L}_r=0,\\
(\ad e_r)^3.\m{L}_{-r}\subseteq(\ad e_r)^2.\m{H}=0,\\
(\ad e_r)^{q+1}.\m{L}_s=\m{L}_{(q+1)r+s}=0,
\end{gather*}

where \(q\) is as described in <ref>.

Consider the following:

Let \(\m{L}\) be a Lie algebra over a field of characteristic 0 and \(\delta\in\Der\m{L}\) be nilpotent, i.e. \(\delta^n=0\) for some \(n\). Then the map \(\exp\delta\), defined as

\begin{equation}\label{exp}
\exp\delta\coloneqq1+\delta+\frac{\delta^2}{2!}+\cdots+\frac{\delta^{n-1}}{(n-1)!},
\end{equation}

is an automorphism of \(\m{L}\).

Let \(k=\mathbb{C}\) first, and take \(\zeta\in\mathbb{C}\). Since \(\ad(\zeta e_r)=\zeta\ad e_r\) is a nilpotent derivation,

\begin{equation}\label{xrz}
x_r(\zeta)\coloneqq\exp(\zeta\ad e_r)
\end{equation}

is an automorphism of \(\mathcal{L}\). It can be shown that \(x_r(\zeta)\) transforms each element of the Chevalley basis into a linear combination of basis
elements, where the coefficients are non-negative integral powers of \(\zeta\) with rational integer coefficients. This fact allows us to define automorphisms of the same type as (<ref>) over any field, by using elements of the corresponding field, instead of \(\mathbb{C}\); this is described for example in <cit.>.

The Chevalley group \(\m{L}(k)\) of type \(\m{L}\) over the field \(k\) is defined as the group of automorphisms of \(\mathcal{L}\) generated by the elements \(x_r(t)\) for \(r\in\Phi\), \(t\in k\), that act on the Chevalley basis for \(\m{L}\) as follows:

\begin{equation}\label{xrt}
\begin{aligned}
x_r(t).h_s&=h_s-A_{s,r}te_r,\quad s\in\Pi,\\
x_r(t).e_s&=\sum_{i=0}^qM_{r,s,i}t^ie_{ir+s},\quad s\ne\pm r,
\end{aligned}
\end{equation}

where \(p,q\) are taken to form an \(r\text{-chain}\) of roots through \(s\), and

\[M_{r,s,i}\coloneqq\frac{1}{i!}N_{r,s}N_{r,r+s}\ldots N_{r,(i-1)r+s}=\pm\frac{(p+1)(p+2)\ldots(p+i)}{i!}=\pm\binom{p+i}{i},\]

where the sign depends on the extraspecial pairs, as before.

Note that the last two relations of (<ref>) can be obtained directly from a Chevalley basis, i.e. without prior knowledge of the structure constants, using only (<ref>) and (<ref>) by:

\begin{equation}\label{xrths}
\end{equation}

\begin{equation}\label{xrtes}
x_r(t).e_s=e_s+t[e_r,e_s]+\frac{1}{2!}[e_r,[e_r,e_s]]+\cdots+\frac{1}{q!}\overbrace{[e_r,[e_r,\ldots[e_r,e_s]\ldots]]}^{q\text{ commutators}}.
\end{equation}

Observe also that we do not need all the roots to generate \(\aut\m{L}\), only \(\Pi\cup-\Pi\), for \(\Pi\) a set of fundamental roots. Indeed, say \(r\) is not a fundamental root and has extraspecial pair \(r=r_1+r_2\), then we have

\[\ad e_r=\ad[e_{r_1},e_{r_2}]=\ad e_{r_1}.\ad e_{r_2}-\ad e_{r_2}.\ad e_{r_1}.\]

Out of the several properties that these generators have, we are interested in the relation between generators corresponding to opposite roots, that is of the form \(x_r(\cdot)\) and \(x_{-r}(\cdot)\). It can be shown that there is a homomorphism \(\phi_r\colon\SL_2(k)\rightarrow\gen{X_r,X_{-r}}\le\m{L}(k)\) for which we have

\[\begin{pmatrix}1&t\\0&1\end{pmatrix}\mapsto x_r(t),\qquad\begin{pmatrix}1&0\\t&1\end{pmatrix}\mapsto x_{-r}(t);\]

see <cit.> for \(k=\mathbb{C}\), and <cit.> for an arbitrary field \(k\).

An important role is played by the elements

\[h_r(\lambda)=\phi_r\begin{pmatrix}\lambda&0\\0&\lambda^{-1}\end{pmatrix},\qquad n_r=\phi_r\begin{pmatrix}0&1\\-1&0\end{pmatrix},\]

where \(\lambda\in k^{\times}\). They are a semisimple element and an involution of \(\PSL_2(k)\), respectively.

In particular, we have that \(h_r(\lambda)\) acts on the Chevalley basis as

\begin{equation}\label{hrl}
\begin{aligned}
h_r(\lambda).h_s&=h_s, &s\in\Pi,\\
h_r(\lambda).e_s&=\lambda^{A_{r,s}}e_s, &s\in\Phi,
\end{aligned}
\end{equation}

while  \(n_r\) acts as

\begin{equation}\label{nr}
\begin{aligned}
n_r.h_s&=h_{w_r(s)},\quad s\in\Pi,\\
\end{aligned}
\end{equation}

where \(w_r(s)\) is given by (<ref>), and \(\eta_{r,s}=\pm1\), again depending on a choice of extraspecial pairs.

Define now \(T\coloneqq\gen{h_r(\lambda)\colon r\in\Phi,\lambda\in k^{\times}}\le\m{L}(k)\), and \(N\coloneqq\gen{T,n_r\colon r\in\Phi}\), then we have that \(T\) is a maximal torus of \(\m{L}(k)\), and \(N\) normalises it (<cit.>).

As before, we only need smaller sets to generate \(T\) and \(N\), in particular we can take \(r\in\Pi\), and \(\lambda\) a primitive element of \(k^{\times}\).

Observe that by <ref> we have the following:

Using the above notation, \(N=T\rtimes W\), where \(W\) is the Weyl group of \(\Phi\).

We will denote by \(h(\lambda_1,\ldots,\lambda_l)\) the product \(h_1(\lambda_1)\ldots h_l(\lambda_l)\), where \(\lambda_i\in k^{\times}\) and \(1\le i\le l\) indexes a set of fundamental roots \(\Pi\).

Chevalley involutions
An important role in our work is played by Chevalley involutions:

Let \(\bb{\G}\) a reductive algebraic group over an algebraically closed field \(k\), and let \(\bb{\T}\le\bb{\G}\) be a maximal torus. A Chevalley involution is an involutive automorphism of \(\bb{\G}\) that stabilises \(\bb{\T}\) and acts as \(t\mapsto t^{-1}\) on \(\bb{\T}\).

Let \(\sigma\) be a Steinberg endomorphism of \(\bb{\G}\), a Chevalley involution of \(\bb{\G}^{\sigma}\) is the restriction of a Chevalley involution of \(\bb{\G}\) that is defined over \(k^{\sigma}\).

Let \(\m{L}\) be a Lie algebra over a field \(k\), finite or algebraic, and let \(G\) be the Chevalley group of \(\m{L}\) as in <ref>; then a Chevalley involution of \(G\) is also an automorphism of \(\m{L}\).

Let \(\varepsilon\coloneqq\left\{e_r,h_s\colon r\in\Phi,s\in\Pi\right\}\) be a Chevalley basis of \(\m{L}\), then the map \(\varphi\) defined by:

\begin{equation}\label{involution}
\begin{aligned}
\iota.e_r&=-e_{-r},\\
\iota.h_s&=-h_s,
\end{aligned}
\end{equation}

is a Chevalley involution.

It is clear that \(\iota\) is an involution and has the required action on \(H=\gen{h_r(\lambda):r\in\Phi,\lambda\in k^{\times}}\), therefore is a Chevalley involution.

Assume for simplicity that \(\varepsilon\) has structure constants \(N_{r,s}=\pm(p+1)\), then we can check that \(\iota\) preserves the relations described in <ref>, so that \(\iota\) is an automorphism of \(\m{L}\) hence lies in \(G\):

* \([\iota.h_r,\iota.h_s]=[-h_r,-h_s]=[h_r,h_s]=0\), for \(r,s\in\Pi\).

* \([\iota.e_r,\iota.e_{-r}]=[-e_{-r},-e_r]=-[e_r,e_{-r}]=-h_r=\iota.h_r\), for \(r\in\Phi\).

* \([\iota.e_r,\iota.e_s]=[-e_{-r},-e_{-s}]=[e_{-r},e_{-s}]=0\), for \(r,s\in\Phi\), \(0\ne r+s\not\in\Phi\).

* \([\iota.h_r,\iota.e_s]=[-h_r,-e_{-s}]=[h_r,e_{-s}]=A_{r,-s}e_{-s}=(-A_{r,s})e_{-s}=A_{r,s}\iota.e_s\), for \(r\in\Pi,s\in\Phi\).

* \([\iota.e_r,\iota.e_s]=[-e_{-r},-e_{-s}]=[e_{-r},e_{-s}]=N_{-r,-s}e_{-r-s}=-N_{r,s}e_{-r-s}=N_{r,s}\iota.e_{r+s}\), for \(r,s\in\Phi\), \(r+s\in\Phi\).

In (iii) we used \(r+s\in\Phi\Leftrightarrow-r-s\in\Phi\), in (iv) <ref>, and in (v) \(N_{r,s}N_{-r,-s}=-(p+1)^2, N_{r,s}=\pm(p+1)\Rightarrow N_{-r,-s}=\mp(p+1)=-N_{r,s}\).

Note that a similar automorphism could be constructed for other choices of structure constants, but it would need a different coefficient on \(\iota.e_r\) for each \(r\), in order to satisfy (v).

Root data and isogeny

Let \(\bb{\G}\) be a reductive algebraic group. Since all maximal tori are conjugate by <ref>, let \(\bb{\T}\) be any maximal torus of \(\bb{\G}\), and let \(X\coloneqq X(\bb{\T})\) and \(Y\coloneqq Y(\bb{\T})\) be the groups of characters and cocharacters of \(\bb{\T}\), respectively.

Let \(\bb{\T}\) be a torus with character group \(X\) and cocharacter group \(Y\). The map \(\gen{-,-}:X\times Y\rightarrow\mathbb{Z}\), defined by \(\chi(\gamma(t))=\gen{\chi,\gamma}(t)\) for all \(t\in\bb{\T}\), is a perfect pairing between \(X\) and \(Y\) that is linear in each variable; that is, any homomorphism \(X\rightarrow\mathbb{Z}\) is of the form \(\chi\mapsto\gen{\chi,\gamma}\) for some \(\gamma\in Y\), and similarly for \(Y\). Therefore there are group isomorphisms \(Y\simeq\hom (X,\mathbb{Z})\) and \(X\simeq\hom(Y,\mathbb{Z})\).

Let \(\mathfrak{g}\) be the Lie algebra of \(\bb{\G}\), and let \(\ad\) be the adjoint representation described in <ref>. For \(\chi\in X(\bb{\T})\), define
\[\mathfrak{g}_{\chi}\coloneqq\Set{v\in\mathfrak{g}|(\ad t).v=\chi(t)v\;\;\forall t\in \bb{\T}}.\]

The set
\[\Phi(\bb{\G})\coloneqq\Set{\chi\in X(\bb{\T})|\chi\ne0,\mathfrak{g}_{\chi}\ne0}\]
is called the set of roots of \(\bb{\G}\) with respect to \(\bb{\T}\), and the group \(W\coloneqq N_{\bb{\G}}(\bb{\T})/C_{\bb{\G}}(\bb{\T})\) is the Weyl group of \(\bb{\G}\).

\(W\) acts on \(X\) and \(Y\) via \((w.\chi)(t)\coloneqq \chi(t^w)\) for all \(w\in W, \chi\in X, t\in\bb{\T}\), and \((w.\gamma)(c)\coloneqq\gamma(c)^{w^{-1}}\) for all \(w\in W, \gamma\in Y, c\in\bb{\G}_m\), respectively. These actions are faithful, and compatible with \(\gen{-,-}\) (<cit.>).

We can also define a set of coroots \(\Phi^{\vee}\) by associating to \(\alpha\in\Phi\) an element \(\alpha^{\vee}\in Y\) such that \(w_{\alpha}.\chi=\chi-\gen{\chi,\alpha^{\vee}}\alpha\) for all \(\chi\in X\). In particular, there is a unique choice for \(\alpha^{\vee}\), and \(\gen{\alpha,\alpha^{\vee}}=2\) (<cit.>).

Therefore, we can associate to \(\bb{\T}\) a finite set of roots \(\Phi\subset X\) and a set of coroots \(\Phi^{\vee}\subset Y\). We may identify \(X\) and \(Y\) with subgroups of \(E\coloneqq X\otimes_{\mathbb{Z}}\mathbb{R}\) and \(E^{\vee}\coloneqq Y\otimes_{\mathbb{Z}}\mathbb{R}\), and extend the action of \(W\) on these vector spaces.

We can also define the following structure:

A quadruple \((X,\Phi,Y,\Phi^{\vee})\) is called a root datum if

* \(X\simeq Y\simeq\mathbb{Z}^n\), with a perfect pairing \(\gen{-,-}\colon X\times Y\rightarrow\mathbb{Z}\) as in <ref>.

* \(\Phi\subseteq X\), \(\Phi^{\vee}\subseteq Y\) are root systems in \(\mathbb{Z}\Phi\otimes_{\mathbb{Z}}\mathbb{R}\) and \(\mathbb{Z}\Phi^{\vee}\otimes_{\mathbb{Z}}\mathbb{R}\), respectively.

* There exists a bijection \(\alpha\mapsto\alpha^{\vee}\) between \(\Phi\) and \(\Phi^{\vee}\) such that \(\gen{\alpha,\alpha^{\vee}}=2\).

* The reflections \(w_{\alpha}\) of \(\Phi\) are defined as:

\begin{align*}
w_{\alpha}.\chi&=\chi-\gen{\chi,\alpha^{\vee}}\alpha\qquad\forall\chi\in X,\\
w_{\alpha^{\vee}}.\gamma&=\gamma-\gen{\alpha,\gamma}\alpha^{\vee}\qquad\forall\gamma\in Y.
\end{align*}

Observe that given a root datum as above, then the Weyl groups of \(\Phi\) and \(\Phi^{\vee}\) are isomorphic via the map \(w_{\alpha}\mapsto w_{\alpha^{\vee}}\).

As we can associate a Lie algebra \(\mathcal{L}\) to a root system \(\Phi\), we can associate the group related to \(\mathcal{L}\) to a root datum:

Let \(\Phi\) be the root system of a connected reductive group \(\bb{\G}\) with respect to the maximal torus \(\bb{\T}\), and set \(\Phi^{\vee}\coloneqq\set{\alpha^{\vee}|\alpha\in\Phi}\). Then \((X(\bb{\T}),\Phi,Y(\bb{\T}),\Phi^{\vee})\) is a root datum, where \(X,Y\) denote the set of characters and cocharacters of \(\bb{\T}\), respectively.

We then have the following:

Two semisimple linear algebraic groups are isomorphic if and only if they have isomorphic root data. For each root datum there exists a semisimple algebraic group which realises it. Such a group is simple if and only if its root system is indecomposable.

Recall from <ref> that \(X\simeq\hom(Y,\mathbb{Z})\), and that \(\mathbb{Z}\Phi\subseteq X\), \(\mathbb{Z}\Phi^{\vee}\subseteq Y\). So by taking the (injective) natural homomorphism \(\hom(Y,\mathbb{Z})\rightarrow\hom(\mathbb{Z}\Phi^{\vee},\mathbb{Z})\eqqcolon\Omega\) induced by restriction, we may identify \(\mathbb{Z}\Phi\) as a subgroup of \(\Omega\) via \(\mathbb{Z}\Phi\subseteq X\subseteq\Omega\).

The fundamental group of \(\Phi\) is \(\Lambda=\Lambda(\Phi)\coloneqq\Omega/\mathbb{Z}\Phi\); observe that it does not depend on \(X\).

Let \(\bb{\G}\) be a semisimple algebraic group with root datum \((X,\Phi,Y,\Phi^{\vee})\). Then \(\Lambda(\bb{\G})\coloneqq\Omega/X\), with \(\Omega\) defined as above, is called the fundamental group of \(\bb{\G}\).

\(\bb{\G}\) is simply connected if \(X=\Omega\), i.e. \(\Lambda(\bb{\G})=1\).

\(\bb{\G}\) is of adjoint type if \(X=\mathbb{Z}\Phi\).

We denote the algebraic group \(\bb{\G}\) with root system \(\Phi\) by \(\bb{\G}_{\ad}\) or \(\bb{\G}_{\sc}\) when \(\Phi\) is of adjoint or simply connected type, respectively.

A homomorphism \(\varphi\colon\bb{\G}\rightarrow\bb{\G}_1\) of algebraic groups is called an isogeny if it is surjective and has finite kernel. If such a morphism exists, \(\bb{\G}\) and \(\bb{\G}_1\) are isogenous.

Simply connected and adjoint groups are related by isogenies due to the following result:

Let \(\bb{\G}\) be a semisimple algebraic group over an algebraically closed field \(k\) of characteristic \(p\), with root system \(\Phi\). Then there exist natural isogenies \(\bb{\G}_{\sc}\stackrel{\pi_1}{\rightarrow}\bb{\G}\stackrel{\pi_2}{\rightarrow}\bb{\G}_{\ad}\), where \(\bb{\G}_{\sc}\) is a simply connected group and \(\bb{\G}_{\ad}\) an adjoint group both with root system \(\Phi\), \(\ker\pi_1\simeq\Lambda(\bb{\G})_{p'}\), \(\ker\pi_2\simeq(\Lambda(\bb{\G}_{\ad})/\Lambda(\bb{\G}))_{p'}\), and both \(d\pi_1\) and \(d\pi_2\) are isomorphisms.

Here, \(\bb{\G}_p'\) denotes the \(p'\text{-part}\) of \(\bb{\G}\), that is the subgroup of \(\bb{\G}\) obtained by taking the elements of \(\bb{\G}\) of order coprime with \(p\).

* If \(\Phi\) is the root system \(A_{n-1}\), it has fundamental group the cyclic group of order \(n\); \(\bb{\G}_{\sc}\) and \(\bb{\G}_{\ad}\) are the groups \(\SL_n\) and \(\PGL_n\), respectively. The corresponding finite Chevalley groups obtained by taking the points over \(\mathbb{F}_q\) are \(\SL_n(q)\) and \(\PGL_n(q)\), respectively; furthermore there is a third group, the simple group \(\PSL_n(q)\), which is the image of the natural map from \(\SL_n(q)\le\GL_n(q)\) to \(\PGL_{n}(q)\).

* The group \(\bb{\E}_7\), denoted by its root system \(\E_7\), has fundamental group the cyclic group of order 2, with two isogeny types, \(\bb{\E}_{7,\sc}\) and \(\bb{\E}_{7,\ad}\). Similarly to the case of \(\A_n\), there are three different finite Chevalley groups over \(\mathbb{F}_q\) associated to it: the simple group \(\E_7(q)\), its Schur cover \(\E_{7,\sc}(q)\), and a subgroup \(\E_{7,\ad}(q)\) of its automorphism group. If \(q\) is even, they all coincide, while if \(q\) is odd we have that \(\E_7(q)\) has index 2 in \(\E_{7,\ad}(q)\), and \(\E_7(q)\) is the quotient of \(\E_{7,\sc}(q)\) over its centre (of order 2).

* Other groups, like those with root system \(\F_4\) or \(\E_8\), have a trivial fundamental group, resulting in only one isogeny type, and one corresponding finite Chevalley group.

Centralisers of semisimple elements
We will need to construct the centraliser of a given semisimple element of \(\bb{\G}\), using the following result:

Let \(\bb{\G}\) be a connected reductive group, \(s\) a semisimple element of \(\bb{\G}\), and \(\bb{\T}\) a maximal torus of \(\bb{\G}\) containing \(s\). Then

* \(C_{\bb{\G}}(s)^{\circ}=\gen{\bb{\T},\bb{\X}_{\alpha}\colon\alpha(s)=1}\).

* \(C_{\bb{\G}}(s)=\gen{\bb{\T},\bb{\X}_{\alpha},n_w\colon\alpha(s)=1,s^w=s}\).

Here, \(n_w\) denotes a representative in \(N\) of the element \(w\) of the Weyl group \(W=N/H\), as in <ref>; \(\bb{\G}^{\circ}\) denotes the connected component of \(\bb{\G}\) containing the identity element of \(\bb{\G}\); \(\bb{\X}_{\alpha}\) denotes the root subgroup of \(\alpha\) with respect to the torus \(\bb{\T}\), as described in <ref>; \(\alpha(s)=1\) means that the root space \(\mathcal{L}_{\alpha}\) is fixed by \(s\); \(s^w=s\) means that \(w\in W(\bb{\T})\) stabilises the set of roots whose root space is fixed by \(s\).

In particular, (i) means that we can construct a connected component of the centraliser of \(s\) by taking a torus and the root subgroups corresponding to the root spaces that are eigenspaces under the action of \(s\). Note that we can obtain them by using the relations (<ref>) and (<ref>), respectively.

Let \(\bb{\G}\) be a connected reductive group whose derived group \(\bb{\G}'\) is simply-connected. Let \(s\) be a semisimple element of \(\bb{\G}\). Then \(C_{\bb{\G}}(s)\) is connected.

By combining <ref> and <ref>, we obtain the following:

Let \(\bb{\G}\) be a connected reductive group whose derived group \(\bb{\G}'\) is simply-connected, \(s\) a semisimple element of \(\bb{\G}\), and \(\bb{\T}\) a maximal torus of \(\bb{\G}\) containing \(s\). Then \(C_{\bb{\G}}(s)=\gen{\bb{\T},\bb{\X}_{\alpha}\colon\alpha(s)=1}\).

Another takeaway from <ref> is that the structure of \(C_{\bb{\G}}(s)\) for a semisimple element \(s\) of \(\bb{\G}\) depends entirely on the fixed-point space of the action of \(s\) on the Lie algebra associated to \(\bb{\G}\).

The fact that the smallest centraliser of \(s\) is a maximal torus is connected to the following:

Let \(\bb{\G}\) be a connected reductive group, \(s\) a semisimple element of \(\bb{\G}\), and \(\m{L}\) the Lie algebra associated to \(\bb{\G}\). Then the fixed point space \(\m{L}^s\) contains a Cartan subalgebra of \(\m{L}\).

Let \(\bb{\T}\) be a maximal torus of \(\bb{\G}\) containing \(s\), then \(\bb{\T}\) is simultaneously diagonalisable in the adjoint representation of \(\bb{\G}\) and stabilises a Cartan decomposition \(\m{H}\oplus\bigoplus_{\alpha\in\Phi}\m{L}_{\alpha}\) of \(\m{L}\), where \(\Phi\) is the set of roots associated to \(\bb{\T}\) (as in <ref>).

Since \(s\in\bb{\T}\), \(s\) acts on \(\m{L}\) as described in (<ref>); in particular, it has eigenvalues 1 on \(\m{H}\).

An immediate consequence is the following:

Let \(\bb{\G}\) be a connected reductive group, \(s\) a semisimple element of \(\bb{\G}\), and \(\m{L}\) the Lie algebra of rank \(l\) associated to \(\bb{\G}\). Then \(\dim\m{L}^s\ge l\); if equality holds, \(\m{L}^s\) is a Cartan subalgebra of \(\m{L}\).

The argument used in <ref> can be used to show that the result holds for any subgroup of a maximal torus.

A presentation of PSL(2,q)

We will use a presentation of the group \(\PSL_2(q)\), \(q=p^r\), that has three generators \(u,s,t\) based on the matrices

\begin{equation}\label{matrices}
u=\begin{pmatrix}1&1\\0&1\end{pmatrix},\quad s=\begin{pmatrix}\omega^{-1}&0\\0&\omega\end{pmatrix},\quad t=\begin{pmatrix}0&1\\-1&0\end{pmatrix},
\end{equation}

where \(\omega\) is a primitive element of \(\mathbb{F}_q^{\times}\). Note that (<ref>) generate \(\SL_2(q)\), but we will keep calling \(u,s,t\) the corresponding elements in the quotient modulo scalar matrices.

Observe that \(u\) and \(s\) generate a Borel subgroup of \(\PSL_2(q)\).

In order to describe the relations of the presentation, we need the following notation: for any polynomial \(g(X)=\sum_{i=0}^eg_iX^i\in\mathbb{Z}[X]\), where \(0\le g_i<p\) for \(0\le i\le e\) so that \(g\) can be seen as an element of \(\mathbb{F}_p[X]\), define

\begin{align*}
&=a^{g_0}b^{-1}a^{g_1}b^{-1}a^{g_2}\ldots b^{-1}a^{g^e}b^e,
\end{align*}

where \(a,b\) are elements of a given group, in this case \(\gen{u,s,t}\).

Let \(k,l\in\mathbb{Z}\) be such that \(\omega^{2k}=\omega^{2l}+1\) and \(\mathbb{F}_q=\mathbb{F}_p[\omega^{2k}]\), and define \(\delta=(k,l)\) (the GCD of \(k\) and \(l\)).

Let \(m(X)\in\mathbb{F}_p[X]\) be the minimal polynomial of \(\omega^{2\delta}\); if \(\gamma\in\mathbb{F}_q\), let \(g_{\gamma}(X)\in\mathbb{F}_p[X]\) be such that \(g_{\gamma}(\omega^{2\delta})=\gamma\) and \(\deg g_{\gamma}<\deg m\).

If \(q>9\) is odd, the group \(\PSL_2(q)\) is isomorphic to a group with generators \(u,s,t\) and relations:

\begin{equation}\label{PSLrelations}
\begin{aligned}
\text{\emph{(i)} }&u^p=1.\\
\text{\emph{(ii)} }&u^{s^k}=uu^{s^l}=u^{s^l}u.\\
\text{\emph{(iii)} }&[[u^{m(X)}]]_{s^{\delta}}=1.\\
\text{\emph{(iv)} }&u^s=[[u^{g_{\omega^2}(X)}]]_{s^{\delta}}.\\
\text{\emph{(v)} }&t^2=1.\\
\text{\emph{(vi)} }&s^t=s^{-1}.\\
\text{\emph{(vii)} }&t=uu^tu.\\
\text{\emph{(viii)} }&st=[[u^{g_{\omega^{-1}}(X)}]]_{s^{\delta}}[[u^{g_{\omega}(X)}]]_{s^{\delta}}^t[[u^{g_{\omega^{-1}}(X)}]]_{s^{\delta}}.
\end{aligned}
\end{equation}

* While it may not be obvious from the relations, one can deduce that \(s\) has order \((q-1)/(2,q-1)\); see <cit.>.

* By (v), we can replace (vi) with \((ts)^2=1\), and (vii) with \((tu)^3=1\).

* In <cit.> it is shown that there exist choices of \(\omega\) such that \(\delta\le2\).

* If \(\delta=1\) then (iv) can be removed; see <ref> below.

* If \(q=p\) then \(\omega\in\mathbb{F}_p\), so \(m(X)=X-\omega^{2\delta}\), and \(g_{\omega^i}(X)=\omega^i\) for all \(i\in\mathbb{Z}\).

In the setting of <ref>, if \(\delta=1\) then the relation (iv) is redundant and can be removed.

If \(\deg m(X)=1\), that means \(\omega^2\in\mathbb{F}_p\) and \(m(X)=X-\omega^2\), so we can take \(g_{\omega^2}(X)=\omega^2\). Therefore \([[u^{g_{\omega^2}(X)}]]_s=u^{\omega^2}\) and \([[u^{m(X)}]]_s=u^{-\omega^2}s^{-1}us\), hence (iii) and (iv) are both \(u^s=u^{\omega^2}\).

If \(\deg m(X)>1\), then we can take \(g_{\omega^2}(X)=X\), so \([[u^{g_{\omega^2}(X)}]]_s=s^{-1}us\), meaning that (iv) becomes \(u^s=u^s\) and can be removed.

Let us write explicit relations for the groups we will study: \(\PSL_2(25)\), \(\PSL_2(27)\), \(\PSL_2(37)\), \(\PSL_2(39)\); we will always choose \(w\in\mathbb{F}_q\) such that \(\delta=1\) when possible, to apply <ref>.

The group \(\PSL_2(25)\) is isomorphic to a group with generators \(u,s,t\) and relations

\begin{equation}\label{25presentation}
\begin{aligned}
\text{\emph{(i)} }&u^5=1.\\
\text{\emph{(ii)} }&u^{s^2}=uu^{s^4}=u^{s^4}u.\\
\text{\emph{(iii)} }&us^{-2}u^4s^{-2}us^4=1.\\
\text{\emph{(iv)} }&u^s=u^2s^{-2}u^3s^2.\\
\text{\emph{(v)} }&t^2=1.\\
\text{\emph{(vi)} }&(ts)^2=1.\\
\text{\emph{(vii)} }&(tu)^3=1.\\
\text{\emph{(viii)} }&st=us^{-2}us^2(u^4s^{-2}u^3s^2)^tus^{-2}us^2.
\end{aligned}
\end{equation}

Let \(\omega\) be a primitive element of \(\mathbb{F}_{25}\) with minimal polynomial \(X^2+4X+2\); then \(\omega^4=\omega^8+1\) so \(k=2\), \(l=4\), \(\delta=2\). The minimal polynomial of \(\omega^{2\delta}=\omega^4\) is \(m(X)=X^2+4X+1\) (observe that \(m(X)\) is independent on the choice of \(\omega\)), and we can take \(g_{\omega}(X)=3X+4\), \(g_{\omega^{-1}}(X)=X+1\), \(g_{\omega^2}(X)=3X+2\). The result follows from <ref>.

Since in this case we have \(\delta=2\), we will compute an additional relation between \(u\) and \(s\) that we will use in <ref>.

If \(u,s,t\) satisfy the presentation (<ref>), then
\begin{equation}\label{25extrarelation}u^4s^{-1}u^3s^{-1}us^2=1.\end{equation}

Unfortunately, we did not manage prove (<ref>) directly from (<ref>); however, it can be verified with  or by using the matrices in (<ref>), since their images under the quotient map \(\SL_2(25)\rightarrow\PSL_2(25)\) satisfy (<ref>).

Observe that this relation is similar to (iii) of (<ref>), since it can be written as \([[u^{\mu(X)}]]_s=1\), where \(\mu(X)\) is the minimal polynomial of \(\omega^2\).

The group \(\PSL_2(27)\) is isomorphic to a group with generators \(u,s,t\) and relations

\begin{equation}\label{27presentation}
\begin{aligned}
\text{\emph{(i)} }&u^3=1.\\
\text{\emph{(ii)} }&u^s=uu^{s^6}=u^{s^6}u.\\
\text{\emph{(iii)} }&u^2s^{-1}us^{-1}us^{-1}us^3=1.\\
\text{\emph{(iv)} }&t^2=1.\\
\text{\emph{(v)} }&(ts)^2=1.\\
\text{\emph{(vi)} }&(tu)^3=1.\\
\text{\emph{(vii)} }&st=us^{-1}u^2s(s^{-1}us^{-1}u^2s^2)^tus^{-1}u^2s.
\end{aligned}
\end{equation}

Let \(\omega\) be a primitive element of \(\mathbb{F}_{27}\) with minimal polynomial \(X^3+2X+1\); then \(\omega^2=\omega^{12}+1\) so \(k=1\), \(l=6\), \(\delta=1\). The minimal polynomial of \(\omega^{2\delta}=\omega^2\) is \(m(X)=X^3+X^2+X+2\), and we can take \(g_{\omega}(X)=2X^2+X\) and \(g_{\omega^{-1}}(X)=2X+1\). The result follows from <ref> and <ref>.

The group \(\PSL_2(37)\) is isomorphic to a group with generators \(u,s,t\) and relations

\begin{align*}
\text{\emph{(i)} }&u^{37}=1.\\
\text{\emph{(ii)} }&u^s=uu^{s^{13}}=u^{s^{13}}u.\\
\text{\emph{(iii)} }&u^s=u^4.\\
\text{\emph{(iv)} }&t^2=1.\stepcounter{equation}\tag{\theequation}\label{37presentation}\\
\text{\emph{(v)} }&(ts)^2=1.\\
\text{\emph{(vi)} }&(tu)^3=1.\\
\text{\emph{(vii)} }&st=u^{19}tu^2tu^{19}.
\end{align*}

Let \(\omega=2\), thought as an element of \(\mathbb{F}_{37}\); then \(\omega^2=\omega^{26}+1\) so \(k=1\), \(l=13\), \(\delta=1\). Then \(m(X)=X-4\), \(g_2(X)=2\), and \(g_{2^{-1}}(X)=2^{-1}=19\). The result follows from <ref> and <ref>.

The group \(\PSL_2(29)\) is isomorphic to a group with generators \(u,s,t\) and relations

\begin{equation}\label{29presentation}
\begin{aligned}
\text{\emph{(i)} }&u^{29}=1.\\
\text{\emph{(ii)} }&u^{s^{11}}=uu^s=u^su.\\
\text{\emph{(iii)} }&u^s=u^4.\\
\text{\emph{(iv)} }&t^2=1.\\
\text{\emph{(v)} }&(ts)^2=1.\\
\text{\emph{(vi)} }&(tu)^3=1.\\
\text{\emph{(vii)} }&st=u^{15}tu^2tu^{15}.
\end{aligned}
\end{equation}

Let \(\omega=2\), thought as an element of \(\mathbb{F}_{29}\); then \(\omega^{22}=\omega^2+1\) so \(k=11\), \(l=1\), \(\delta=1\). Then \(m(X)=X-4\), \(g_2(X)=2\), and \(g_{2^{-1}}(X)=2^{-1}=15\). The result follows from <ref> and <ref>.

By checking with , we observed that (ii) is redundant in the cases we study and it may be removed.

Locating a Borel subgroup of PSL(2,q) in exceptional groups
We need to know where a Borel subgroup of \(\PSL_2(q)\) lies when embedded in an algebraic group, for which we use the following results.

Let \(K\) be a supersoluble subgroup of a connected reductive Lie group \(\bb{\G}\). Then there exists a maximal torus \(\bb{\T}\) of \(\bb{\G}\) such that \(K\le N_{\bb{\G}}(\bb{\T})\).

Furthermore, let \(\sigma\) be a Frobenius endomorphism of \(\bb{\G}\). If each \(K_i\) in a normal series for \(K\)  is \(\sigma\text{-stable}\), then \(\bb{\T}\) can be taken to be \(\sigma\text{-stable}\).

The first part is <cit.>, the additional condition about \(\sigma\text{-stability}\) is proved for example in <cit.>.

Let \(E\) be an elementary abelian subgroup of an algebraic group \(\bb{\G}\) in odd characteristic:

* If \((\bb{\G},E)=(\bb{\F}_4,2^2)\), then there exists a maximal torus of \(\bb{\G}\) containing \(E\).

* If \((\bb{\G},E)=(\bb{\F}_4,5^2)\), then there exists a maximal torus of \(\bb{\G}\) containing \(E\).

* If \((\bb{\G},E)=(\bb{\F}_4,3^3)\), then \(E\) is not contained in a maximal torus, and its normaliser in \(\bb{\G}\) is \(3^3\rtimes\SL_3(3)\).

This follows from <cit.>, where the result is studied for all choices of \(\bb{\G}\) and elementary abelian subgroups \(E\).

Note that a similar statement for finite exceptional groups can be found in <cit.>.

Counting conjugacy classes in the finite and algebraic group
Part of our work consists in constructing a Borel subgroup \(B\) of the group \(H\simeq\PSL_2(p^r)\) that we are trying to embed in \(\bb{\G}\) and \(\G\), and counting how many copies of \(H\) contain it.

If there is more than one subgroup isomorphic to \(H\) containing the same \(B\), then we need to check whether they are conjugate or not, but that will be dealt with case by case.

Since we will work in odd characteristic, we begin by observing that \(B\simeq p^r\rtimes\frac{p^r-1}{2}\), so \(B\) has socle \(B_0=B'\simeq p^r\), an elementary abelian \(p\text{-subgroup}\).

Let \(\bb{\G}\) be a linear algebraic group, an element \(x\in\bb{\G}\) is called regular if \(\dim C_{\bb{\G}}(x)\) is the smallest possible among all the elements of \(\bb{\G}\).

Let \(\bb{\G}\) be a connected reductive group, \(s\in\bb{\G}\) a semisimple element, and \(\bb{\T}\) a maximal torus of \(\bb{\G}\) containing \(s\) with associated root system \(\Phi\). The following are equivalent:

* \(s\) is regular.

* \(\alpha(s)\ne1\) for all \(\alpha\in\Phi\).

* \(C_{\bb{\G}}(s)^{\circ}=\bb{\T}\).

Here, the notation is the same as in <ref>.

We say that a subgroup \(K\le\bb{\G}\) is regular if it is contained in a maximal torus \(\bb{\T}\) of \(\bb{\G}\) and there is no \(\alpha\in\Phi\) such that \(\alpha(s)=1\) for all \(s\in K\), where \(\Phi\) is the root system associated to \(\bb{\T}\).

In particular, from <ref> it follows that \(C_{\bb{\G}}(K)^{\circ}=\bb{\T}\).

When \(H\) is isomorphic to one of \(\PSL_2(25)\), \(\PSL_2(37)\), \(\PSL_2(39)\), it will follow from <ref> and <ref> that \(B_0\) is always contained in a maximal torus \(\bb{\T}\).

Suppose there are \(B_1,B_2\simeq B\) contained in \(\bb{\G}\), such that \(B_1',B_2'\simeq B_0\) are \(\bb{\G}\text{-conjugate}\), with \(B_i=B_i'\rtimes w_i\). In particular, we may assume \(B_1'=B_2'=B_1\cap B_2=B_0\subset\bb{\T}\). If for \(i=1,2\) \(B_i'\) is regular and \(C_{\bb{\T}}(w_i)\) is finite, then \(B_1,B_2\) are \(\bb{\G}\text{-conjugate}\).

Since \(B_0\) is regular, \(C_{\bb{\G}}^{\circ}(B_0)=\bb{\T}\) for some maximal torus \(\bb{\T}\supset B_0\).

It is straightforward to prove that \(N_{\bb{\G}}(B_0)\subseteq N_{\bb{\G}}(C_{\bb{\G}}^{\circ}(B_0))\): let \(g\in N_{\bb{\G}}(B_0)\), \(b\in B_0\), and \(c\in C_{\bb{\G}}^{\circ}(B_0)\), then \(b^{c^g}=(b^{g^{-1}})^{cg}=(b^{g^{-1}})^g=b,\) that is \(c^g\in C_{\bb{\G}}^{\circ}(B_0)\).

This implies that \(B_1,B_2\subset N_{\bb{\G}}(B_0)\subseteq N_{\bb{\G}}(\bb{\T})\), therefore we can take \(B\) to lie in the normaliser of a maximal torus; using <ref> this means that \(B=B_0\rtimes w\) where \(w\) is the preimage in \(N_{\bb{\G}}(\bb{\T})\) of an element of \(W(\bb{\G})\).

Let \(\tau_w\) be the endomorphism of \(\bb{\T}\) induced by the conjugation action of \(w\in N_{\bb{G}}(\bb{\T})\). Since \(C_{\bb{\T}}(w)\) is finite, by <ref> \(\tau\) is a Steinberg endomorphism. Then, by <ref>, the map \(t\mapsto\tau_w(t)t^{-1}=t^wt^{-1}\) is surjective, i.e. for any \(t_1\in\bb{\T}\) there is \(t\in\bb{\T}\) s.t. \(t_1=t^wt^{-1}\). Rearranging the terms gives \(wt_1=w^{t^{-1}}\), so the elements of \(w\bb{\T}\) are all \(\bb{\T}\text{-conjugate}\); in particular, \(B_1\) and \(B_2\) are \(\bb{\T}\text{-conjugate}\) as for the cases we are considering there is a unique conjugacy class in \(W(\bb{\G})\) of elements of order \(\size{w}\), so for both groups the element extending \(B_0\) lies in \(\bb{\T}w\).

Therefore, \(B\) is unique up to \(\bb{\T}\text{-conjugacy}\).

For the finite case, let \(\T=\bb{\T}^{\sigma}\).

There are at most \(\size{C_{\T}(w)}\) conjugacy classes of complements to \(\T\) in \(Tw\).

This time we use the map \(\varphi_w\colon t\mapsto[t,w]\); clearly, \(\varphi_w\colon\T\rightarrow\T\), and it is a homomorphism since for any \(t,u\in\T\) we have:

\begin{align*}
\varphi_w(tu)&=[tu,w]=\\
\end{align*}

where the last step follows because \(u,[t,w]\in\T\), which is abelian.

Observe now that given \(s\in\T\), by rearranging \((tw)^s=uw\) we obtain \(t^{-1}u=[s,w]\), which means that \(tw\) and \(uw\) are \(\T\text{-conjugate}\) iff \(t^{-1}u\in[\T,w]=\im\varphi_w\).

Therefore, \(\T w\) splits into \(\size{\ker\varphi_w}=\size{C_{\T}(w)}\) conjugacy classes, not all of which have elements with the same order as \(w\).

While in general not all the classes have elements with the same order as \(w\), they do for example if \(C_{\bb{\T}}(w)=Z(\bb{\G})\), since multiplication by elements of the centre does not change the order.

Let \(\bb{\G}\) be an algebraic group with Steinberg endomorphism \(\sigma\), and \(H\le \G=\bb{\G}^{\sigma}\); we want to know whether the \(\bb{\G}\text{-conjugates}\) of \(H\) contained in \(\G\) are \(\G\text{-conjugate}\) or they split into multiple conjugacy classes.

Let \(G\) be an abstract group and \(\sigma\) be an automorphism of \(G\). We say that \(g_1,g_2\in G\) are \(\sigma\text{-conjugate}\) if there exists \(x\in G\) such that \(g_2=x^{-1}g_1\sigma(x)\). It can be shown that \(\sigma\text{-conjugacy}\) is an equivalence relation, and its equivalence classes are called \(\sigma\text{-conjugacy}\) classes of \(G\).

Let \(\bb{\G}\) be a linear algebraic group with Steinberg endomorphism \(\sigma\), and let \(\bb{\G}_1\) be a closed connected normal \(\sigma\text{-stable}\) subgroup. Then the quotient map induces a natural bijection from \(\sigma\text{-conjugacy}\) classes of \(\bb{\G}\) to \(\sigma\text{-conjugacy}\) classes of \(\bb{\G}/\bb{\G}_1\).

Let \(\bb{\G}\) be a connected linear algebraic group with a Steinberg endomorphism \(\sigma\) acting transitively on a non-empty set \(V\) with a compatible \(\sigma\text{-action}\), i.e. \(\sigma(g.v)=\sigma(g).\sigma(v)\) for all \(g\in \bb{\G}\), \(v\in V\). Then:

* \(\sigma\) has fixed points on \(V\), i.e. \(V^{\sigma}\ne\emptyset\).

* If the stabilizer \(\bb{\G}_x\) is closed for some \(x\in V\), then for any \(v\in V^{\sigma}\) there is a natural 1-to-1 correspondence:
\[\Set{\bb{\G}^{\sigma}\text{-orbits on }V^{\sigma}}\leftrightarrow\Set{\sigma\text{-conjugacy classes in }\bb{\G}_v/\bb{\G}_v^{\circ}}.\]

See for example <cit.>.

Let \(H\) be a subgroup of \(\G=\bb{\G}^{\sigma}\). If \(C_{\bb{\G}}(H)\) is connected, then all \(\bb{\G}\text{-conjugates}\) of \(H\) that lie in \(\G\) are conjugate in \(\G\). If \(C_{\bb{\G}}(H)\le\G\), then the \(\bb{\G}\text{-conjugates}\) of \(H\) that lie in \(\G\) split into a number of conjugacy classes equal to the number of conjugacy classes of elements of \(C_{\bb{\G}}(H)\).

Let \(H=\gen{h_1,\ldots,h_n}\) be a subgroup of \(\G=\bb{\G}^{\sigma}\), and apply theorem <ref> by taking \(V\) to be \(\Set{(h_1^g,\ldots,h_n^g)\colon g\in\bb{\G}}\). By construction, \(\bb{\G}\) acts transitively on \(V\) by conjugation, and since we have \(H\le\G\) then \(\sigma(h_i)=h_i\), implying that \(\sigma(h_i^g)=h_i^{\sigma(g)}\) for any \(i=1,\ldots,n\) and \(g\in\bb{\G}\), i.e. \(\sigma\) is compatible with the action of \(\bb{\G}\) on \(V\).

We also have that \(\bb{\G}_{(h_1,\ldots,h_n)}=C_{\bb{\G}}(H)\) is closed since \(C_{\bb{\G}}(h_1)\cap\ldots\cap C_{\bb{\G}}(h_n)\) is an intersection of centralisers of elements, which are closed sets, so by <ref> we have a 1-to-1 correspondence between the \(\G\text{-orbits}\) on \(V^{\sigma}\), i.e. the number of conjugacy classes of \(H\) in \(\G\), and the number of \(\sigma\text{-conjugacy}\) classes in \(\bb{\G}_{(h_1,\ldots,h_n)}/\bb{\G}_{(h_1,\ldots,h_n)}^{\circ}=C_{\bb{\G}}(H)/C_{\bb{\G}}(H)^{\circ}\).

If \(C_{\bb{\G}}(H)\) is connected, then \(C_{\bb{\G}}(H)/C_{\bb{\G}}(H)^{\circ}\) is trivial so there is only one such conjugacy class.

If \(C_{\bb{\G}}(H)\le\G\), since \(\sigma\) fixes elements of \(\G\) then the \(\sigma\text{-classes}\) on \(C_{\bb{\G}}(H)\) are the \(\bb{\G}\text{-conjugacy}\) classes.

Since we normally start by counting classes in \(\bb{\G}\) and then use that knowledge to move down to the finite group \(\G\), we need to know whether \(H\) embeds in \(\G\) in the first place. In some cases this depends on the irrationalities in the characters of \(H\), i.e. whether the ground field splits a certain polynomial.

The field \(\mathbb{F}_q\) for odd \(q\) is a splitting field for the polynomial

* \(X^2-X-1\);

* \(X^2-X-7\);

* \(X^2-X-9\);

* \(X^3-X^2-2X+1\);

if and only if

* \(q\equiv0,\pm1\bmod5\);

* \(q\equiv0,\pm1,\pm4,\pm5,\pm6,\pm7,\pm9,\pm13\bmod29\);

* \(q\equiv0,\pm1,\pm3,\pm4,\pm7,\pm9,\pm10,\pm11,\pm12,\pm16\bmod37\);

* \(q\equiv0,\pm1\bmod7\).

The first three cases can be shown by computing the roots of the corresponding polynomial:

* \((1\pm\sqrt{5})/2\);

* \((1\pm\sqrt{29})/2\);

* \((1\pm\sqrt{37})/2\).

The splitting field must contain \(\sqrt{d}\), \(d=5,29,37\), respectively. If \(d\) is 0 in \(\mathbb{F}_q\) then the polynomial splits as \(q\) is odd. The other congruences follow by using quadratic reciprocity, as \(d\) is a square in \(\mathbb{F}_q\) if and only if \(q\) is a square in \(\mathbb{F}_d\), and the list of congruences can be computed explicitly.

For (iv), if \(q\equiv0\bmod7\) then \(X^3-X^2-2X+1\) factorises as \((X+2)^3\). Otherwise, observe that \(-\zeta_7-\zeta_7^{-1}\) is a root, with \(\zeta_7\) a 7th root of unity:

\begin{align*}
\left.X^3-X^2-2X+1\right|_{X=-\zeta_7-\zeta_7^{-1}}&=(-\zeta_7-\zeta_7^{-1})^3-(-\zeta_7-\zeta_7^{-1})^2-2(-\zeta_7-\zeta_7^{-1})+1\\
\end{align*}

Similarly, one can verify that the other roots are \(-\zeta_7^2-\zeta_7^{-2}\) and \(-\zeta_7^3-\zeta_7^{-3}\).

If \(q\equiv1\bmod7\), then \(\mathbb{F}_q\) admits 7th roots of unity, so the polynomial splits. Otherwise, consider \(\mathbb{F}_{q^a}\) with \(q^a\equiv1\bmod7\), and observe that the roots are invariant under the map \(x\mapsto x^q\), i.e. they lie in \(\mathbb{F}_q\), iff \(q\equiv\pm1\bmod7\).

Another key result that allows us to count conjugacy classes in an algebraic group is the following lifting lemma, proved by Larsen:

If \(H\) is a finite group and \(\bb{\G}(\cdot)\) is a split group scheme, then the cardinality of the finite set \(\hom(H,\bb{\G}(k))/\bb{\G}(k)\) is independent of the algebraically closed field \(k\) of characteristic not dividing \(\size{H}\).

Here, \(\hom(A,B)\) denotes the the set of group homomorphism from \(A\) to \(B\), and \(\hom(A,B)/B\) is the set of \(B\text{-conjugacy}\) classes of such homomorphisms.

This essentially tells us that to count the different conjugacy classes of embeddings of \(H\) in \(\bb{\G}\) in characteristic \(p\notdivides\size{H}\), we can arbitrarily choose one such \(p\), and the result automatically generalises.

Since we only use schemes indirectly through this lemma, we refer to <cit.> for a detailed overview of the subject. In particular, we have that algebraic groups are group schemes, and we can think of \(\bb{\G}(\cdot)\) as the map associating the algebraically closed field \(k\) to the algebraic group \(\bb{\G}(k)\), where \(\G\) is a root datum. As we study embeddings in \(\bb{\F}_4\) and \(\bb{\E}_{7,\ad}\), which are split reductive groups, we are able to use this result.

Maximal subgroups of exceptional groups of Lie type

In this section we will focus only on the groups of exceptional type, and present some well-known results about the classification of their maximal subgroups. When some groups belong to a known list we will often omit it, as it may be a rather long one, and refer to the original result for those interested.

Let \(\bb{\G}\) be a simple algebraic group of exceptional adjoint type over an algebraically closed field \(k\) of characteristic \(p\), and let \(\aut\bb{\G}\) be the abstract group generated by inner, graph, diagonal, and field automorphisms of \(\bb{\G}\). The maximal closed subgroups of positive dimension in \(\bb{\G}\) are given by the following:

Let \(\bb{\G}_1\) be a group satisfying \(\bb{\G}\le\bb{\G}_1\le\aut\bb{\G}\). Let \(\bb{\X}\) be a proper closed connected subgroup of \(\bb{\G}\) which is maximal among proper connected closed \(N_{\bb{\G}_1}(\bb{X})\text{-invariant}\) subgroups of \(\bb{\G}\). Then one of the following holds:

* \(\bb{\X}\) is either parabolic or reductive of maximal rank.

* \(\bb{\G}=\bb{\E}_7\), \(p\ne2\), and \(N_{\bb{\G}}(\bb{\X})=(2^2\times\bb{\D}_4).\sym_3\).

* \(\bb{\G}=\bb{\E}_8\), \(p\ne2,3,5\), and \(N_{\bb{\G}}(\bb{\X})=\bb{\A}_1\times\sym_5\).

* \(\bb{\X}\) belongs to a list of known cases.

The subgroups \(\bb{\X}\) in (ii), (iii), and (iv) exist, are unique up to conjugacy in \(\aut\bb{\G}\), and are maximal among closed connected \(N_{\bb{\G}}(\bb{\X})\text{-invariant}\) subgroups of \(\bb{\G}\).

For a given \(\bb{\G}\), let \(\sigma\) be a Steinberg endomorphism of \(\bb{\G}\), and set \(\G=\bb{\G}^{\sigma}\); we denote by \(\overline{\G}\) an almost simple group with socle \(\G/Z(\G)\).

In particular, we are interested in the cases when \(\G\) is one of \(\F_4(q)\), \(\E_6(q)\), \(\!\prescript{2}{}\E_6(q)\), \(\E_7(q)\), and \(\E_8(q)\), as the maximal subgroups of the other possible \(\G\) have been completely classified.

Let \(\mathscr{X}\) be the set of the maximal positive-dimensional subgroups of \(\bb{\G}\) given by <ref>; then we denote by \(\mathscr{X}^{\sigma}\) the set of maximal subgroups arising from positive-dimensional subgroups of \(\bb{\G}\), which means:

* \(\mathscr{X}^{\sigma}\) is the set of fixed points \(\bb{\X}^{\sigma}\), where \(\bb{\X}\in\mathscr{X}\) is \(\sigma\text{-stable}\).

* If \(Z(\G)\ne1\), \(\mathscr{X}^{\sigma}\) is the set of images of elements in (i) modulo the centre.

* If we are considering \(\overline{\G}\), \(\mathscr{X}^{\sigma}\) is the set of \(N_{\overline{\G}}(\X)\), where \(\X\) lies in the set in (i) associated to \(\G\).

Observe first that the maximal subgroups of \(\overline{\G}\) split into three categories:

* \(M\cap\G/Z(\G)\) is a maximal subgroup of \(\G/Z(\G)\).

* \(M\cap\G/Z(\G)\) is not a maximal subgroup of \(\G/Z(\G)\). \(M\) is called a novelty maximal subgroup in this case.

* \((\G/Z(\G))\le M\).

In <ref> we will deal only with maximal subgroups of \(\G/Z(\G)\), so we can compute \(M\) by taking their normaliser in \(\overline{\G}\).

The maximal subgroups of \(\overline{\G}\) are described by <cit.>:

Let \(M\) be a maximal subgroup of \(\overline{\G}\), then one of the following holds:

* \(M\in\mathscr{X}^{\sigma}\).

* \(M\) has the same type as \(\bb{\G}\), possibly twisted. For example, \(\E_6(p)\) and \(\!\prescript{2}{}\E_6(p^2)\) are subgroups of \(\E_6(p^2)\) that belong to this category.

* \(\G=\E_8\), \(p>5\), and \(M=(\alt_5\times\alt_6).2^2\).

* \(M\) is an exotic \(r\text{-local}\) subgroup, i.e. the normaliser of an elementary abelian \(r\text{-group}\) for \(r\ne p\).

* \(H=F^*(M)\) is simple.

The subgroup in (iii) was discovered by Borovik in <cit.>, and is unique up to conjugacy; the subgroups in (iv) are known, and a list is given in <cit.>.

The groups in (v) divide naturally in two classes, depending on whether the simple group \(H\) is a group of Lie type in characteristic \(p\), or any other type of simple group; the two classes are called generic and non-generic subgroups, respectively.

For the generic subgroups, we have the following result:

Let \(H=H(q_0)\) be a simple group of Lie type in characteristic \(p\) as defined in <ref>(v). Assume that either of the following holds:

* \(q>u(G)\cdot(2,p-1)\) and \(H(q_0)\) is one of \(\A_1(q_0)\), \(\!\prescript{2}{}B_2(q_0)\), \(\!\prescript{2}{}G_2(q_0)\).

* \(q>9\), \(H(q_0)\) is not as in (i), and \(H(q_0)\ne\A_2^{\epsilon}(16)\).

Then \(H\) satisfies <ref>(i) or (ii).

Here, \(u(G)\) is as in the following table.

\(\G_2\)   \(\F_4\)   \(\E_6\)   \(\E_7\)   \(\E_8\)
\(u(G)\)   12   68   124   388   1312

It may be possible that there are no generic subgroups that do not satisfy <ref>(i) or (ii).

For the non-generic subgroups, the list of potential candidates is given in <cit.>, where the problem of finding an explicit embedding of \(H\) in \(\G\) (and \(\bb{\G}\)) up to conjugacy is still open for some cases.

The subgroups \(H\) are also categorised depending on whether they are primitive subgroups or not, see <ref>. The list of candidate non-generic \(H\) is studied in <cit.>, where they are classified depending on whether they embed in \(\bb{\G}\) as a strongly imprimitive subgroup, as a primitive subgroup, or they may admit embeddings of either type. The complete list is given in <cit.>.

In particular, <cit.> gives a list of triples \((\bb{\G},H,p)\) such that \(H\) embeds in \(\bb{\G}\) in characteristic \(p\) only as a strongly imprimitive subgroup. This has a strong implication for maximality in finite groups:

If \(H\) embeds in \(\bb{\G}\) in characteristic \(p\) only as a strongly imprimitive subgroup, then \(\overline{\G}\) has no maximal subgroup with socle \(H\).

Therefore, primitive subgroups are the only ones with a chance of being the socle of a maximal subgroups of \(\overline{\G}\):

If \(H\) embeds primitively in \(\bb{\G}\), then either \(N_{\overline{\G}}(H)\) is a maximal subgroup of \(\G\) or it is contained in a subgroup \(M\) such that one of <ref>(ii), (iii), (v), holds. If (v) holds, then \(F^*(M)\) admits a primitive embedding in \(\bb{\G}\).

This follows directly from the fact that \(H\) is a simple primitive group.

The complete list of cases studied in <cit.> is rather long, so we will list only those we can attempt with the technique hereby explained:

\(G\)   \(H\)
\(\F_4\)   [c]\(\alt(5),\alt(6),\PSL_2(q)\)
\(q=7,8,13,17,25,27\)
\(\E_6\)   [c]\(\alt(5),\alt(6),\PSL_2(q)\)
\(q=7,8,11,13,17,25,27\)
\(\E_7\)   [c]\(\alt(5),\alt(6),\PSL_2(q)\)
\(q=7,8,11,13,17,19,25,27,29,37\)
\(\E_8\)   [c]\(\alt(5),\alt(6),\PSL_2(q)\)
\(q=7,8,11,13,16,17,19,25,27,29,31,32,41,49,61\)

Observe that we have \(\alt(5)\simeq\PSL_2(4)\simeq\PSL_2(5)\) and \(\alt(6)\simeq\PSL_2(9)\), so we can consider them of Lie type.

As we will see later on, unfortunately the technique we use fails on most of these, generally when \(q\) is “too small”, when doing the steps described in <ref> and <ref>.

We now list some results about these cases that have already been established. In particular, <ref> allows us to ignore the case \(\alt(5)\) altogether, as the group is always strongly imprimitive. We will also specify whether the number of conjugacy classes of embeddings is known.

Embedding of alternating groups
Using the notation that was previously defined in <ref>, we have the following results.

Let \(\G=\bb{\G}^{\sigma}\) be a simple group of type \(\F_4,\E_6,\prescript{2}{}{\E_6},\E_7\), or \(\E_8\). If \(H\le\G\) with \(F^*(H)\simeq\alt(n)\), for \(n=5\) or \(n\ge8\), then \(H\) lies inside a member of \(\mathscr{X}^{\sigma}\).

Let \(\G=\bb{\G}^{\sigma}\) be a simple group of type \(\F_4,\E_6,\prescript{2}{}{\E_6},\E_7\), or \(\E_8\) over a field of characteristic \(p\), and let \(H\le\G\) not lie inside a member of \(\mathscr{X}^{\sigma}\).

* if \(F^*(H)\simeq\alt(6)\), then \((\G,p)\) is one of \((\F_4,3)\), \((\F_4,\ge7)\), \((\E_6,\ge7)\), \((\E_7,\ge5)\), \((\E_8,2)\), \((\E_8,3)\), \((\E_8,\ge7)\);

* if \(F^*(H)\simeq\alt(7)\), then \((\G,p)\) is one of \((\E_7,5)\), \((\E_8,3)\), \((\E_8,5)\), \((\E_8,7)\), \((\E_8,\ge11)\).

Note that in (i), the case \((\F_4,3)\) has been dealt with by the same author, who proved that it may be excluded from the list, although the result has not been published yet at the time of writing.

Therefore, for the alternating groups, only for \(n=6,7\) there is no complete answer yet. We will deal with the case of \(\alt(6)<\bb{\F}_4,\bb{\E}_6\) in characteristic 0 or coprime to \(\size{\alt_6}\) in <ref>.

Embedding of linear groups
These results will be useful to check that our code works; in particular, we can provide an independent confirmation of <ref>, <ref>, and the embedding of \(\PSL_2(61)\) in \(\bb{\E}_8(\mathbb{C})\) of <ref>. The supplementary material includes the computations required in such cases.

Let \(k\) be an algebraically closed field of characteristic that does not divide \(\size{\PSL_2(41)}\). There are three conjugacy classes of \(\PSL_2(41)\text{-subgroups}\) in \(\bb{\E}_8(k)\), the subgroups give rise to six conjugacy classes of embeddings into \(\bb{\E}_8(k)\).

The groups \(\SL_2(41)\) and \(\PGL_2(41)\) are not embedded in \(\bb{\E}_8(\mathbb{C})\).

Let \(k\) be an algebraically closed field of characteristic that does not divide \(\size{\PSL_2(49)}\). There are two conjugacy classes of \(\PSL_2(49)\text{-subgroups}\) in \(\E_8(k)\), the subgroups give rise to four conjugacy classes of embeddings into \(\bb{\E}_8(k)\).

The groups \(\SL_2(49)\) and \(\PGL_2(49)\) are not embedded in \(\bb{\E}_8(\mathbb{C})\).

Kostant's conjecture holds, that is the group \(\PSL_2(q)\) embeds in \(\bb{\G}(\mathbb{C})\) when \((\bb{\G},q)\) is one of \((\bb{\G}_2,13)\), \((\bb{\F}_4,25)\), \((\bb{\E}_6,25)\), \((\bb{\E}_7,37)\), \((\bb{\E}_8,61)\).

For \(\bb{\G}=\bb{\G}_2,\bb{\E}_8\), there is a unique \(\bb{\G}\text{-conjugacy}\) class of subgroups isomorphic to \(\PSL_2(q)\). A similar result is only conjectured for \(\bb{\F}_4\), \(\bb{\E}_6\), and \(\bb{\E}_7\).

Let \(k\) be an algebraically closed field of characteristic that does not divide \(\size{\PSL_2(31)}\). Then there are three conjugacy classes of \(\PGL_2(31)\text{-subgroups}\) in \(\bb{\E}_8(k)\), the subgroups give rise to three conjugacy classes of embeddings into \(\bb{\E}_8(k)\).

This implies the following:

Under the same hypothesis, there are at least three conjugacy classes of \(\PSL_2(31)\text{-subgroups}\) in \(\bb{\E}_8(k)\).

In particular, it is not known whether there are other embeddings of \(\PSL_2(31)\) that do not extend to \(\PGL_2(31)\).

The trilinear form for E6
The contents of this section can be found in <cit.> and <cit.>.

Let \(V\) be a finite-dimensional vector space over a field \(k\). Given \(m\in\mathbb{Z}^+\), denote by \(M_m(V)\) the \(k\text{-space}\) of functions \(V^m\rightarrow k\), i.e. the space of \(m\text{-forms}\). Then \(g\in\GL(V)\) acts on \(\alpha\in M_m(V)\) via
\[(g.\alpha)(v_1,\ldots,v_m)=\alpha(v_1\cdot g,\ldots,v_m\cdot g),\]
where \(v_1,\ldots,v_m\in V\).

An element \(g\in\GL(V)\) is a similarity of \(\alpha\in M_m(V)\) if \(g.\alpha=\lambda_g\alpha\) for some \(\lambda_g\in k^{\times}\). A similarity is an isometry if \(\lambda_g=1\). It is clear that isometries of a given \(\alpha\in M_m(V)\) form a group, called the isometry group of \(\alpha\).

Let \(\alpha\in M_m(V)\) be a symmetric \(m\text{-linear}\) form, and let \(e=(e_1,\ldots,e_n)\) be a basis of \(V\). Then \(\alpha\) is uniquely determined by \(\alpha(e_{i_1},\ldots,e_{i_m})\eqqcolon\alpha_{i_1\ldots i_m}\), where \(1\le i_1,\ldots,i_m\le n\), not necessarily distinct, so that
\begin{equation}\label{mform}\alpha=\sum_{1\le i_1,\ldots i_m\le n}a_{i_1\ldots i_m}e_{i_1}\ldots e_{i_m}.\end{equation}

Each summand of (<ref>) is called a monomial of \(\alpha\), thus \(\alpha\) is uniquely determined by its monomials.

For the rest of the section, we assume that the field \(k\) has characteristic \(\ne2,3\).

A triple \(\mathcal{F}(T,P,f)\) is a 3-form if

* \(f\) is a trilinear form on \(V\).

* \(P:V\times V\rightarrow k\) is linear in the first component and satisfies

\begin{align*}
\end{align*}

for all \(x,y,z\in V\), \(a\in k\).

* \(T:V\rightarrow k\) satisfies:

\begin{align*}
\end{align*}

for all \(x,y\in V\), \(a\in k\).

If \(\mathcal{F}(T,P,f)\) is a 3-form, then \(f\) is symmetric.

Using the definition of \(T\) and \(P\) we have that:

\begin{align*}
\end{align*}

where the brackets are simply to separate the computation of each summands.

\begin{equation}\label{fsym}f(x,y,z)=T(x+y+z)-\sum_{a\in\left\{x,y,z\right\}}T(a)-\sum_{\substack{a\ne b \\ a,b\in\left\{x,y,z\right\}}}P(a,b)\end{equation}

is symmetric.

Since \(\ch k\ne2,3\), then \(P(x,y)=\frac{1}{2}f(x,y,y)\) and \(T(x)=\frac{1}{6}f(x,x,x)\).

Therefore, if we define \(P_x(y)\coloneqq P(x,y)\), then \(P_x\) is a quadratic form, whose associated bilinear form is \(\frac{1}{2}f_x\), where \(f_x\coloneqq f(x,-,-)\).

The first equality follows from the definition of \(P\), since
\[4P(x,y)=P(x,2y)=P(x,y)+P(x,y)+f(x,y,y),\]

so if \(\ch k\ne2\) then by rearranging the terms we obtain the result.

The second equality follows from (<ref>) and the definition of \(T\). In particular, we have that \(T(3x)=27T(x)\), and \(T(2x)=2T(x)+2P(x,x)\), i.e. \(P(x,x)=3T(x)\) if \(\ch k\ne2\), so from (<ref>) we have

\begin{align*}
\end{align*}

and the result follows since \(\ch k\ne2,3\).

Let \(v\in V\), \(U\subseteq V\). We denote by \(x\Delta\) the radical of \(f_x\), and \(U\Delta\coloneqq\bigcap_{x\in U}x\Delta\). We also define \(U\Theta\coloneqq\Set{v\in V|P_v(u)=0\;\;\forall\,u\in U}\).

If \(U\subseteq U\Delta\) we call \(U\) singular. If \(U\subseteq U\Theta\) we call \(U\) brilliant.

For any \(U\subseteq V\), \(U\Delta\) and \(U\Theta\) are subspaces of \(V\).

This follows from the definition, since \(f\) is linear and \(P\) is linear in the first component.

Let \(U\le V\) and \(v\in V\), then \(v\in U\Theta\) iff \(f(v,u,u')=0\) for all \(u,u'\in U\).

This follows from the definition of \(P\):

\[f(v,u,u')=P(v,u+u')-P(v,u)-P(v,u').\]

Let \(v\in U\Theta\), then all the terms on the right hand side are zero since \(u,u',u+u'\in U\).

Vice versa, if \(f(v,u,u')=0\) for all \(u,u'\in U\) then \(P(v,u+u')=P(v,u)+P(v,u')\) for all \(u,u'\in U\). In particular, for all \(a\in k, u\in U\) we have that \(aP(v,u)=P(v,au)=a^2P(v,u)\), so \(P_v(u)=0\) for all \(u\in U\), i.e. \(v\in U\Theta\).

Let \(V\) be a \(\dimn{27}\) vector space with basis \(X=\Set{x_i,x_i',x_{ij}|1\le i,j\le6, i<j}\), with the convention \(x_{ij}=-x_{ji}\). Let \(f\) be the symmetric trilinear form whose nonzero monomials are \(x_ix_j'x_{ij}\), \(1\le i\ne j\le 6\) and \(x_{1^d2^d}x_{3^d4^d}x_{5^d6^d}\) where \(d\in C\subset\alt_6\).

\(C\) is a set of coset representatives for \(\alt_{\Delta}\) in \(\alt_6\), where \(\alt_{\Delta}\) denotes the stabiliser in \(\alt_6\) of the partition \(\Delta=\left\{\left\{1,2\right\},\left\{3,4\right\},\left\{5,6\right\}\right\}\) of \(\left\{1,\ldots,6\right\}\). For a choice of \(C\), the images of \((1,2;3,4;5,6)\) are as follows:

12 34 56   13 24 65   14 23 56   15 26 43   16 25 34

12 35 64   13 26 54   14 25 63   15 24 36   16 23 45

12 36 45   13 25 46   14 26 35   15 23 64   16 24 53

For \((i,j;k,l;m,n)\) as in the above table, the monomial \(x_{ij}x_{kl}x_{mn}\) has sign \(+1\) in \(f\).

The Dickson 3-form is the 3-form \((T,P,f)\), where \(f\) is the trilinear form above, which we call the \(\E_6\text{-form}\).

Since we are working with \(\ch k\ne 2,3\), \(P\) and \(T\) are \(P(x,y)=\frac{1}{2}f(x,y,y)\), \(T(x)=\frac{1}{6}f(x,x,x)\), by <ref>.

The isometry group of the Dickson 3-form is the universal Chevalley group of type \(\E_6\).

Using the notation of <ref>, let
\[V_9:=\gen{x_{16},x_{25},x_{34},x_{12},x_{56},x_2',x_5,x_1,x_6'},\]
and denote by \(\mathcal{V}_9\) the orbit of \(V_9\) under the action of \(\E_6\).

A 3-decomposition of \(V\) is a decomposition of \(V\) into \(A_1\oplus A_2\oplus A_3\) such that each of the \(A_i\) is a member of \(\mathcal{V}_9\) and \(A_i\oplus A_j=A_k\Theta\) for \(i,j,k\) distinct.

A property of 3-decompositions we will need is the following:

The stabiliser in \(\E_6\) of a 3-decomposition is isomorphic to the product of a Cartan subgroup of \(\E_6\) with three copies of \(\SL_3(k)\) (i.e. a group of type \(\A_2\A_2\A_2\)) extended by \(\sym_3\).

Recall that there is a canonical correspondence between bilinear forms on \(V\) and linear maps \(V\otimes V\rightarrow k\):

* Given \(f\) a bilinear form on \(V\), the associated map is \(u\otimes v\mapsto f(u,v)\).

* Given a linear map \(F:V\otimes V\rightarrow k\), the associated bilinear form is \(f(u,v)=F(u\otimes v)\).

The set of linear maps \(V\otimes V\rightarrow k\) is the dual space of \(V\otimes V\), i.e. \((V\otimes V)^*\), which is canonically isomorphic to \(V^*\otimes V^*\) since we will only work with a finite-dimensional \(V\), so bilinear forms can be thought as elements of \(V^*\otimes V^*\).

The symmetric square of \(V^*\) is the quotient of \(V^*\otimes V^*\) by the ideal generated by elements of the form \(x\otimes y-y\otimes x\), \(x,y\in V^*\), so by following the maps described above, there is a canonical map between \(\sym^2(V^*)\) and symmetric bilinear forms of \(V\).

The above considerations extend naturally when \(V\) is a \(kG\text{-module}\).

Let \(f\) be a bilinear form on the \(kG\text{-module}\) \(V\), and \(F\in(V\otimes V)^*\) the associated linear map. Then \(f\) is \(G\text{-invariant}\) iff \(F\in{(V\otimes V)^*}^G\).

Let \(f\) be \(G\text{-invariant}\), i.e. \(f(u,v)=f(g.u,g.v)\) for all \(u,v\in V\) and all \(g\in G\). Then

\begin{align*}
g.F(u\otimes v)&=F(g.(u\otimes v))\\
&=F(g.u\otimes g.v)\\
&=F(u\otimes v).
\end{align*}

Vice versa, let \(F\in(V\otimes V)^*\) be fixed by \(G\), i.e. \(g.F=F\) for all \(g\in G\), then

\begin{align*}
f(g.u,g.v)&=F(g.u\otimes g.v)\\
&=F(g.(u\otimes v))\\
&=g.F(u\otimes v)\\
&=F(u\otimes v)\\
\end{align*}

In the same way, it follows from the canonical mappings that a symmetric bilinear form \(f\) on \(V\) is \(G\text{-invariant}\) iff the associated element in \(\sym^2(V^*)\) is fixed by \(G\).

In particular, if \(\dim\sym^2(V^*)=1\), then there is a unique symmetric bilinear form on \(V\), up to scalars; if \(V\) is a \(kG\text{-module}\) and \(\dim\sym^2(V^*)^G=1\), then there is a unique \(G\text{-invariant}\) symmetric bilinear form on \(V\).

The above considerations extend to \(\sym^n(V^*)\) and symmetric \(n\text{-linear}\) forms on \(V\). In particular, a symmetric trilinear form \(f\) on \(V\) corresponds to an element \(F\) of \(\sym^3(V^*)\), and \(f\) is \(G\text{-invariant}\) iff \(F\) is fixed by \(G\).

Let \(G\) be a group over \(k\), \(M_1,\ldots,M_n\) be finite-dimensional \(kG\)-modules, \(\Set{\alpha_{i,j}^c}_{c\in\mathcal{C}_{i,j}}\) be a basis of \(\hom_{kG}(M_i,\sym^2(M_j^*))\), for \(1\le i\ne j\le n\), where
\[\m{C}_{i,j}=\left\{1,\ldots,\dim\hom_{kG}(M_i,\sym^2(M_j^*))\right\},\]
and \(\tau_i\) be the canonical \(kG\text{-isomorphism}\) between \(\sym^2(M_i^*)\) and the symmetric bilinear forms of \(M_i\), for \(1\le i\le n\).

We define \(f_{i,j}^c\colon M_i\times M_j\times M_j\rightarrow k\) to be the \(G\text{-invariant}\) trilinear map defined by \((x,y,y')\mapsto(\tau_j(\alpha^c_{i,j}(x)))(y,y')\), for \(1\le i\ne j\le n\).

We also define \(\Set{f_i^b}_{b\in\mathcal{B}_i}\) to be a basis of the set of \(G\text{-invariant}\) symmetric trilinear forms of \(M_i\), for \(1\le i\le n\), where \(\m{B}_i=\left\{1,\ldots,\dim\sym^3(M_i^*)\right\}\).

We assume that everything is chosen in such a way that all the \(f_{i,j}^c\) and \(f_i^b\) defined as above are nontrivial.

Note that in later chapters, we will often identify \(f_{i,j}^c\) with \(\alpha_{i,j}^c\) for simplicity.

Let \(G\) be a group over \(k\), \(M=M_1\oplus M_2\) a finite-dimensional \(kG\text{-module}\), and \(f\) a \(G\text{-invariant}\) symmetric trilinear form on \(M\). Then

\begin{equation}\label{formdecomposition}f=\sum_{i=1}^2\sum_{b\in\mathcal{B}_i}a_i^bf_i^b+\sum_{i\ne j}\sum_{c\in\mathcal{C}_{i,j}}a_{i,j}^cf_{i,j}^c,\end{equation}

where \(f_i^b,f_{i,j}^c,\m{B}_i,\m{C}_{i,j}\) are as in <ref>, and all \(a_i^b,a_{i,j}^c\) lie in \(k\).

Let \(x,y,z\in M\), with projections \(x_i,y_i,z_i\) over \(M_i\). Then using the linearity of \(f\) we have

\[f(x,y,z)=\sum_{1\le i,j,k\le2}f(x_i,y_j,z_k)=\sum_{1\le i,j,k\le2}\left.f\right|_{M_i\times M_j\times M_k}(x_i,y_j,z_k).\]

The conclusion is self-evident using <ref>, since by symmetry we have four terms: \(\left.f\right|_{M_1}\), \(\left.f\right|_{M_2}\), \(\left.f\right|_{M_1\times M_2\times M_2}\), and \(\left.f\right|_{M_2\times M_1\times M_1}\). Each of the first two terms is a linear combinations of generators of the space of trilinear forms of the corresponding module, while each of the latter two terms is a linear combination of generators of \(\hom_{kG}(M_i,\sym^2(M_j^*))\) for \(i\ne j\), in the way described in <ref>.

Despite the unfriendly notation, the takeaway of the lemma is very intuitive: a \(G\text{-invariant}\) trilinear form on a module \(M=M_1\oplus M_2\), i.e. an element of the \(G\text{-fixed}\) point space of \(\sym^3(M)\), is a linear combination of \(\sym^3(M)\), \(\hom_{kG}(M_1,\sym^2(M_2^*))\), \(\hom_{kG}(M_2,\sym^2(M_1^*))\).

This can be generalised to decompositions with more than two direct summands, say \(M=M\oplus N\oplus O\), but it needs additional maps like \(\hom_{kG}(M,N^*\otimes O^*)\) in order to include the terms originating from \(\left.f\right|_{M\times N\times O}(x_M,y_N,z_O)\), where \(x_M,y_N,z_O\) denote the projection of \(x,y,z\) over \(M,N,O\), respectively.

In later chapters, we will sometimes denote \(\left.f\right|_{M\times N\times O}\) as \(f(M,N,O)\) for simplicity.

Let \(U,V\) be finite-dimensional \(kG\text{-modules}\), and \(f\) a \(G\text{-invariant}\) symmetric trilinear form defined on \(U\oplus V\). If \(\dim\hom_{kG}(V,\sym^2(U^*))=0\), then \(V\le U\Theta\).

Since \(\left.f\right|_{V\times U\times U}\) is a linear combination of the \(\size{\m{C}_{i,j}}\) maps \(f_{i,j}^c\) described in <ref>, if \(\size{\m{C}}=\dim\hom_{kG}(V,\sym^2(U^*))=0\) then \(\left.f\right|_{V\times U\times U}=0\), which means \(V\le U\Theta\) by <ref>.

<ref> describe the constructions used in <ref>.

Let \(\m{L}\) be a Lie algebra of dimension \(d\) and rank \(l\) over a finite field \(k\) of our choice. In <ref> we construct generators of the corresponding Chevalley group \(\G=\G(k)\) in its adjoint representation, described in <ref>, as matrices in \(\GL_d(k)\). We also construct generators of a maximal torus \(T\) and of its normaliser \(N=N_{\G}(T)\). In <ref> we describe an easy way to check whether an element of \(\GL_d(k)\) lies in \(\G\).

Let \(\overline{k}\) be the algebraic closure of \(k\), \(\overline{\mathcal{L}}=\mathcal{L}\otimes_k\overline{k}\), and let \(\bb{\G}\) be the subgroup of \(\GL_d(\overline{k})\) preserving \(\overline{\mathcal{L}}\), in particular \(\G=\bb{\G}^{\sigma}\) for a Frobenius endomorphism \(\sigma\) of \(\bb{\G}\).

Since the underlying vector space of \(\m{L}\) is a \(\G\text{-module}\) under the natural action of \(\G\) in its adjoint representation, we will often denote \(\m{L}\) by \(L(\G)\), the adjoint module of \(\G\). The same holds for \(\overline{\m{L}}\) and \(L(\bb{\G})\). Whether we consider the Lie algebra structure or the module structure will be clear from the context. The minimal module of \(\G\) will be denoted by \(M(\G)\) instead.

We then proceed to construct \(s,u,t\in\bb{\G}\) that satisfy a presentation for \(\PSL_2(q)\), described in <ref>. In <ref> we construct a suitable \(s\in N\) and find \(u\in T\) satisfying the required relations, and in <ref> we describe how to find all the possible choices for \(t\in\bb{\G}\). This is divided in several steps:

* In <ref> we describe how to find a Chevalley basis \(\varepsilon_s\) of \(\m{L}\) that diagonalises \(s\).

* Using \(\varepsilon_s\), in <ref> we construct an involution \(e\in\G\) that inverts \(s\).

* In <ref> we describe how to compute the possible \(c\in\bb{\G}\) such that \(t=ce\).

Observe that the last step is not always possible, depending on \(q\) and \(k\). The embeddings we study in <ref> are some of the cases where no issues arise, and we are able to compute the possible choice of \(t\).

<ref> describe how to construct the LDU decomposition of \(M\in\SL_{n+1}(k)\), if \(M\) admits one, and use it to find the preimage of \(M\) under the standard representation \(\rho:\A_n\rightarrow\SL_{n+1}(k)\). This is used in <ref> to construct a copy of \(\alt_6<\F_4(k)\le\GL_{52}(k)\), for a suitable field \(k\).

Let \(G\) be a group and \(V\) a finite-dimensional \(kG\text{-module}\). In <ref> we describe how to construct \(G\text{-invariant}\) symmetric trilinear forms defined over \(V\), if any exists. This is used in <ref>, where we construct the \(\C_4\text{-invariant}\) form induced by the \(\E_6\text{-invariant}\) form described in <ref>.

Let \(G\) be a finite group, \(V\) a finite-dimensional \(kG\text{-module}\), \(f\) a \(G\text{-invariant}\) symmetric trilinear form on \(V\), and \(H\) a subgroup of \(G\). In <ref> we describe how we can write \(\left.f\right|_V\) in terms of \(\left.f\right|_{V\restri{H}}\). This is used in <ref>, where we use our knowledge of \(\left.f\right|_{V\restri{H}}\) to study \(\left.f\right|_V\).

We will point out when an algorithm or a step of it is already available in  via a built-in function; likewise, we will highlight whenever we use a built-in  function.

Construction of a maximal torus, its normaliser, and the corresponding root subgroups
Let \(\m{L}\) be a Lie algebra of the desired type, with underlying root system \(\Phi=\Phi(\m{L})\), and let \(\varepsilon=(\varepsilon_1,\ldots,\varepsilon_d)\) be a Chevalley basis of \(\m{L}\);  we choose \(\varepsilon\) so that the structure constants are \(N_{s,r}=\pm(p+1)\) for each \(r,s\in\Pi\), \(\Pi\) a base of \(\Phi\), as described in <ref>.

Denote by \(\G\) the group of automorphisms of \(\mathcal{L}\) generated by \(x_{\pm i}(1)\) for \(i=1,\ldots,l\), and let \(T\) and \(N\) be the groups defined in <ref>. We can construct generators for \(\G\), \(T\), and \(N\) by using their action on \(\varepsilon\) described by (<ref>), (<ref>), (<ref>) respectively, in particular we construct \(x_{\pm i}(1), h_i(\lambda), n_i\) where \(\lambda\) is a primitive element of \(k^{\times}\), and \(\alpha_i\) for \(i=1,\ldots,l\) are a set of fundamental roots.

The supplementary material contains a function that constructs \(x_{\pm i}(1), h_i(\lambda), n_i\) for an arbitrary Lie algebra \(\m{L}\) over a finite field \(k\), provided that the given Chevalley basis \(\varepsilon\) is a list sorted as described in <ref>.

There are a few ways we can check that the matrices \(x_{\pm i}(1), h_i(\lambda), n_i\) we construct are correct.

If working with , as mentioned earlier one could verify that this construction matches the built-in version.

Alternatively, one could verify that the group generated is isomorphic to the one we wanted by computing a composition series for it. In  this can be done with the  function, whose algorithm is described in <cit.>.

Naturally, we can perform a membership test, explained in <ref>, to confirm that all the matrices we constructed are in \(\G\); and it is easy but useful to verify that \(n_i\) normalises \(h_j(\lambda)\) for any choice of \(i,j\), and that the matrices \(h_i(\lambda)\) commute with each other.

Membership test
Since we are working with large groups, stored by generating matrices, computations in the group can be quite difficult. Therefore, to check whether a matrix \(g\in\GL_d(k)\) is an element of \(\G\) we use the following result <cit.>:

Let \(\m{L}=\gen{a_1,\ldots,a_I}\) be a Lie algebra, and let \(g\in\GL_{\dim\m{L}}(k)\). Then \(g\in\gen{x_{\pm i}(1)|1\le i\le I}\) iff \(g^{-1}\ad(a_i)g=\ad(a_ig)\) for \(1\le i\le I\).

\[\m{G}\coloneqq\Set{a\in\m{L}|g^{-1}\ad(a)g=\ad(ag)}=\Set{a\in\m{L}|[a,bg^{-1}]g=[ag,b]\;\forall b\in\m{L}}.\]

First, we show that \(\m{G}=\Set{a\in\m{L}|[b,a]g=[bg,ag]\;\forall b\in\m{L}}\).

Let \(a\in\m{L}\) be such that \([a,bg^{-1}]g=[ag,b]\) for all \(b\in\m{L}\), and let \(b\in\m{L}\). Then

\begin{align*}
[b,a]g	&=-[b,a]g\\
\end{align*}

Vice versa, let \(a\in\m{L}\) be such that \([b,a]g=[bg,ag]\) for all \(b\in\m{L}\), and let \(b\in\m{L}\). Then

\begin{align*}
[ag,b]	&=-[b,ag]\\
\end{align*}

If we show that \(\m{G}\) is a subalgebra of \(\m{L}\), then \(\m{G}=\m{L}\) because \(a_1,\ldots,a_I\in\m{G}\) are generators for \(\m{L}\). Therefore, \(g\) preserves the Lie product on \(\m{L}\) and \(g\in\aut\m{L}\).

By <ref>, we only need to show that \(\m{G}\) is closed under Lie multiplication. Let \(a,a'\in\m{G}\), then for any \(b\in\m{L}\) we have

\begin{align*}
[b,[a,a']]g	&\eqtext{J}[[a',b],a]g+[[b,a],a']g\\
\end{align*}

therefore \([a,a']\in\m{G}\). Here, J denotes using the Jacobi identity on \(\m{L}\), and G denotes using the fact that \(a,a'\in\m{G}\).

Therefore, our membership test is performed as follows:

* Select a set \(a_1,\ldots,a_I\) of generators for \(\m{L}\) as a Lie algebra. A set of generators for \(\m{L}\) as a vector space also works, but is in general larger thus requires more computations.

* Check whether \(g^{-1}\ad(a_i)g=\ad(a_ig)\) for \(i=1,\ldots,I\).

* \(g\in\G\) if and only if equality holds for all \(i\).

Constructing an element of NG(T) and its action on T

In all the embeddings we study, listed in <ref>, we end up looking for an element \(u\) lying in a maximal torus \(\T\) of \(\G\), and an element \(s\in N=N_{\G}(\T)\) that acts on \(u\) in a certain way. We construct them by taking \(s\in N_{\G}(\T)\) first, for a given \(\T\), then find a corresponding \(u\).

In all cases except \(\PSL_2(27)\), the \(u\) and \(s\) we mention in this section correspond to those of the presentation of the given \(\PSL_2(q)\), \(q\) a power of \(p\), so we will assume this is the case, and \(s,u\) are as described in <ref>.

Recall from <ref> that \(N=\T\rtimes W\), and from <ref> that \(N=\gen{\T,n_i|i=1,\ldots,l}=\gen{h_i(\lambda),n_i|i=1,\ldots,l}\), so we know that a set of generators for \(W\) is given by the images of \(n_1,\ldots,n_l\) via the map \(N\rightarrow N/\T\). We will then take \(s\) to be a random element of \(\widetilde{W}=\gen{n_i|i=1,\ldots,l}\) of order \(\frac{q-1}{2}\).

In all the cases we study, there is a unique conjugacy class in \(W\) of elements of the appropriate order, making this task quite easy, as we can find \(s\) by doing a random search in \(\widetilde{W}\) until we find an element of the correct order, and making sure that its image on \(W\) has that same order. In other cases, we would have to make sure it lies in the correct conjugacy class(es), which can be done on a case-by-case scenario: the trace could be a distinguishing factor, as is simply checking whether a suitable \(u\) exists for the given \(s\).

Working with , usually we can take \(s\) to be a preimage in \(N=N_{\G}(\T)\) of an element \(w\) of the Weyl group \(W=N/\T\) of the required order.

We use \(s\) to find \(u\). Start by taking the maximal torus \(\T\) as constructed in <ref>, and compute its Sylow \(p\text{-subgroup}\) \(\T_p\) by taking \(\frac{\size{k}-1}{p}\text{-th}\) powers of its generators \(h_i(\lambda)\). Now the matrices \(\widehat{h}_i\coloneqq h_i(\lambda)^{\frac{\size{k}-1}{p}}\) have order \(p\) and commute with each other, so \(\T_p\simeq p^l\) is elementary abelian.

We can construct the matrix \(\overline{s}\in\GL_l(p)\) representing the action of \(s\) on \(\T_p\) in the following ways:

* If we have \(\widehat{h}_i\) as matrices in \(\GL_d(k)\), we can assume they are diagonal, \(\widehat{h}_i=\mathrm{diag}(d_{i1},\ldots,d_{id})\). Then, for \(1\le i\le l\), the entries of the \(i\text{-th}\) row of \(\overline{s}\) are given by the solution of the system of equation given by

\begin{equation}\label{sbarsystem1}\overline{s}_{i1}\log_ad_{1j}+\cdots+\overline{s}_{il}\log_ad_{lj}=\log_a((\widehat{h}_i^s)_{jj}),\end{equation}

for \(1\le j\le d\), where \(a=\omega^{\frac{\size{k}-1}{p}}\), \(\omega\) a primitive element of \(k\).

* If we have \(\widehat{h}_i\) as torus elements written in the form \(\widehat{h}_i=h(a^{\delta_{1i}},\ldots,a^{\delta_{li}})\) described in <ref>, where \(\delta_{ij}=1\) if \(i=j\) and 0 otherwise, and \(a=\omega^{\frac{\size{k}-1}{p}}\), with \(\omega\) a primitive element of \(k\), let \(\widehat{h}_i^s=h(b_{i1},\ldots,b_{il})\) for \(i=1,\ldots,l\). Then the entries of the \(i\text{-th}\) row of \(\overline{s}\) are given by \(\overline{s}_{ij}=\log_a(b_{ij})\), \(j=1,\ldots,l\).

In both cases we compute \(\overline{s}_{ij}\) as elements of \(\left\{0,\ldots,p-1\right\}\), then we take the corresponding representative in \(\mathbb{F}_p\).

For (i), we have that since the \(\widehat{h}_i\) are diagonal, and \(s\) normalises the torus they lie in, then \(\widehat{h}^s\) is also diagonal, and \(\widehat{h}_i^s=\widehat{h}_1^{\overline{s}_{i1}}\ldots\widehat{h}_l^{\overline{s}_{il}}\) for some \(\overline{s}_{i1},\ldots,\overline{s}_{il}\in\left\{0,\ldots,p-1\right\}\), \(1\le i\le l\), as \(\widehat{h}_1,\ldots,\widehat{h}_l\) is a basis of \(\T_p\).

We can divide this relation into \(l\) equations, one for each entry of \(\widehat{h}_i^s\), that is
\begin{equation}\label{sbarsystem11}(\widehat{h}^s_i)_{jj}=(d_{1j})^{\overline{s}_{i1}}\ldots(d_{lj})^{\overline{s}_{il}},\end{equation}

for \(j=1,\ldots,d\). Recall that \(\widehat{h}_i\) and \(\widehat{h}_i^s\) are both diagonal matrices in \(\GL_d(k)\), so their diagonal entries are never 0 and we can compute discrete logarithms: let \(\omega\) be a primitive element of \(k\), and let \(a=\omega^{\frac{\size{k}-1}{p}}\), so that \(a\) has order \(p\); then by applying \(\log_a\) to each side of (<ref>) we obtain (<ref>).

Since \(\widehat{h}^s\in\T_p\), the system of equations (<ref>) has a solution and since \(\widehat{h}_1,\ldots,\widehat{h}_l\) is a basis of \(\T_p\), the solution is unique.

For (ii), the setting is different but the argument is similar. Let \(\omega\) be a primitive element of \(k\) and \(a=\omega^{\frac{\size{k}-1}{p}}\), then \(h_1=h(\omega,1,\ldots,1),\ldots,h_l=h(1,\ldots,1,\omega)\) is a basis of \(\T\) (recall that multiplication is component-wise) and \(\widehat{h}_1=h(a,1,\ldots,1),\ldots,h_l=h(1,\ldots,1,a)\), , is a basis of \(\T_p\). Since \(s\) normalise \(\T_p\), for \(i=1,\ldots,l\) we have
\begin{equation}\label{sbarsystem2}\widehat{h}_i^s=\widehat{h}_1^{\overline{s}_{i1}}\ldots\widehat{h}_l^{\overline{s}_{il}}=h(a^{\overline{s}_{i1}},\ldots,a^{\overline{s}_{il}}),\end{equation}
for some \(\overline{s}_{i1},\ldots,\overline{s}_{il}\in\left\{0,\ldots,p-1\right\}\). Therefore by writing \(\widehat{h}_i^s=h(b_{i1},\ldots,b_{il})\) we have \(\overline{s}_{ij}=\log_a(b_{ij})\), for \(j=1,\ldots,l\).

Therefore \(\overline{s}\in\GL_l(p)\) is a matrix representing \(s\).

Recall from <ref> that the action of \(s\) on \(U\coloneqq\gen{u^{s^{\delta}}}\) is described by the relation (iii) of (<ref>).

* Let \(\overline{s}^*\) be the rational canonical form of \(\overline{s}\), and let \(A\in\GL_l(p)\) be such that \(A\overline{s}A^{-1}=\overline{s}^*\). In  this can be computed with the  function, implemented as described in <cit.>.

* Locate the block of \(\overline{s}^*\) whose minimal polynomial is \(m(X)\), the minimal polynomial of \(\omega^{2\delta}\).

* Take any non-zero vector \(v\in\mathbb{F}_p^l\) acted on by said block, and let \(z=(z_i)_{i=1}^l\coloneqq vA\).

* Let \(u\coloneqq\prod_{i=1}^l\widehat{h}_i^{z_i}\in\GL_d(k)\), which corresponds to the adjoint representation of the torus element \(h(z_1,\ldots,z_l)\).

Observe that if \(q=p\) then \(v\) is a vector acted on by the block with minimal polynomial \(m(X)=X-\omega^{2\delta}\), thus \(z\) is simply an eigenvector of \(\overline{s}\) corresponding to the eigenvalue \(\omega^{2\delta}\).

When studying \(\PSL_2(25)\), we are forced to take \(\delta=2\) in (<ref>), therefore we use (<ref>) instead. This means that at step (ii) above we use \(\mu(X)\), the minimal polynomial of \(\omega^2\), instead of \(m(X)\). After computing \(u\), we have to confirm that the relations (iii) and (iv) of (<ref>) are satisfied.

We also discuss here how we check whether \(C_{\bb{\T}}(s)\) is finite, and if so how to compute it, by using the fact that \(s\) is a representative in \(N_{\bb{\G}}(\bb{\T})\) of some element \(w\in W(\bb{\G})\).

Recall from <ref> that \(W\) is generated by the reflections of \(\Phi\), and the action of \(W\) as permutation group over \(\Phi\) corresponds to an action of \(W\) as an automorphism of \(\overline{\m{L}}\), obtained by permuting the root subspaces \(\overline{\m{L}}_{\alpha}\), \(\alpha\in\Phi\). There is a representation of degree \(l\) of \(W\), called reflection representation, which describes the action of \(W\) on a Cartan subalgebra \(\m{H}\) of \(\m{L}\).

Let \(w^*\in N_{\bb{\G}}(\bb{\T})\) be a preimage of \(w\in W\). Then \(w\) does not have an eigenspace with eigenvalue 1 in the reflection representation of \(W\) iff \(C_{\bb{\T}}(w)\) is finite.

This is immediate from the definition of reflection representation. Let \(\m{H}\) be a Cartan subalgebra of \(\overline{\m{L}}\); if \(v\in\m{H}\) then the subspace \(\gen{v}\) is fixed by \(w\) iff \(\gen{\ad v}\le\bb{\T}\) centralises \(w^*\), and \(\gen{\ad v}\) is a 1-dimensional (infinite) torus.

In , if we have \(\G\) as a , then we can obtain \(W\) as a permutation group using the  function. Then,  gives the reflection representation of \(W\).

If we have \(s\in N_{\G}(\T)\), but we do not know the reflection representation of the image of \(s\) in \(W(\G)\), then we can still deduce that \(C_{\bb{\T}}(s)\) is finite if the conjugation action of \(s\) on \(\T\) has no fixed points: if \(C_{\bb{\T}}(s)\) was infinite, it would be a subtorus \({(\overline{k}^{\times})}^j\le{(\overline{k}^{\times})}^l\simeq\bb{\T}\) for \(1\le j\le l\), which then would have nontrivial intersection with \(\T\).

Observe that the opposite is not true in general, as the existence of a (finite) set of fixed points \(\T^*\) on \(\T\) does not imply the existence of infinite set of fixed points on \(\bb{\T}\) containing \(\T^*\).

Let \(s\in N_{\bb{\G}}(\bb{\T})\). If \(C_{\bb{\T}}(s)\) is finite, it can be constructed by computing \(C_{E}(s)\), where \(E\) ranges among the elementary abelian subgroups of \(\bb{\T}\) for the primes dividing the order \(o(s)\) of \(s\).

Let \(A\) be a finite subgroup of \(\bb{\T}\), and let \(A=A_1\times A_2\), where \(\size{A_2}\) is coprime to \(o(s)\) and the set of prime divisors of \(\size{A_1}\) is a subset of the prime divisors of \(o(s)\). Then, by the Schur-Zassenhaus theorem, all complements of \(A_2\) in \(\gen{A_2,s}\) are \(A_2\text{-conjugates}\), hence \(C_A(s)\le A_1\).	Therefore, we may assume \(A=A_1\), and the result follows.

For \(j\ge0\) and for any prime \(p\) dividing \(\size{k}-1\), define \(\T_{p,j}\coloneqq\gen{h_i(\lambda^{\frac{\size{k}-1}{p^j}}):i=1,\ldots,l}\) where \(\lambda\) is a primitive element of \(k\), in particular \(\T_{p,j}\) is a homocyclic subgroup of \(\bb{\T}\) with \(l\) cyclic factors of order \(p^j\), and \(\T_{p,0}\) is the trivial group for all \(p\).

The result of <ref> means that if \(o(s)=\prod_{p\in P}^np^{a_p}\) for some set \(P\) of primes, then for each \(p\in P\) we can find the \(p\text{-part}\) of \(C_{\bb{\T}}(s)\) by computing \(C_{\T_{p,i}}(s)\) for \(i=0,1,\ldots\), and stopping when \(C_{\T_{p,i}}(s)=C_{\T_{p,i-1}}(s)\).

Observe that if the process has not yet stopped when reaching \(i\) such that \(p^i\) does not divide \(\size{k}-1\), then we need to use a field extension \(k^j\) instead, for some \(j>1\). The process will always stop after a finite number of steps since by assumption \(C_{\bb{\T}}(s)\) is finite.

Since we will need to compute \(N_{\G}(\gen{u,s})\), we also mention the following result.

Let \(B\le\G=\bb{\G}^{\sigma}\) and let \(B'\) be the derived subgroup of \(B\). If \(B'\) is supersoluble then \(N_{\G}(B)\le N_{\G}(\T)\) for a maximal torus \(\T\) of \(\G\).

Observe first that \(N_{\G}(B)\le N_{\G}(B')\); indeed, for all \(x,y\in B\) and all \(g\in N_{\G}(B)\) we have
\[[x,y]^g=[x^g,y^g]\in[B,B]=B',\]
so \(g\in N_{\G}(B')\).

Since \(B'\) is supersoluble, then \(N_{\G}(B')\le N_{\G}(T)\) for a maximal torus \(\T\) of \(\G\) by <ref>.

We will apply this result when \(B=\gen{u,s}\) and \(B'\) lies in a maximal torus \(\T\) of \(\G\); in such case we will have that \(B'\) is supersoluble and lies in \(\T\), so we can construct generators of \(N_{\G}(\T)\), and we may compute \(N_{\G}(B)\) by taking \(N_{N_{\G}(\T)}(B)\).

Constructing the final involution t
Given a Lie algebra \(\m{L}\) over \(k\) of dimension \(d\), the corresponding Chevalley groups \(\G(k)\) and \(\bb{\G}(\overline{k})\), and \(u,s\in\G\) as matrices in \(\GL_d(k)\) constructed as in <ref>, we now want to construct the possible \(t\in\bb{\G}\) such that \((u,s,t)\) satisfy the presentation for \(\PSL_2(q)\) given by <ref>.

Observe that the relations \(t^2=(st)^2=1\) imply that each possible \(t\) is an involution of the form \(t=ce\), where \(c\) centralises \(s\), and \(e\) is an involution that inverts \(s\). We construct \(e\in\bb{\G}\) first, then use the relation \((ut)^3=1\) to narrow down our search for \(c\).

If we find any suitable \(c\in\bb{\G}\) and compute the corresponding \(t\), then we only have to confirm that the relation (viii) of (<ref>) is satisfied.

A Chevalley basis relative to a semisimple element
The aim of this section is to describe methods to obtain a Chevalley basis on which a given semisimple element \(s\) acts diagonally. This will be needed in <ref>, where we use a matrix that realises the change of basis to diagonalise \(s\).

We preface this section by mentioning that if we are working with  and we know \(s\) as a word in \(x_{\pm i}(t)\) and \(h_i(\lambda)\), then in all cases but \(\PSL_2(29)\) we could use the  function instead: the second output is an element that conjugates \(s\) into an element of the torus generated by the \(h_i(\lambda)\), so we can use it to map the current Chevalley basis to one that makes \(s\) diagonal.