# Radically Compositional Cognitive Concepts

Toby B. St Clere Smithe
Department of Experimental Psychology
University of Oxford
Oxford, UK
OX2 6NW
<EMAIL_ADDRESS>http://tsmithe.net

###### Abstract

Despite ample evidence that our concepts, our cognitive architecture, and
mathematics itself are all deeply compositional, few models take advantage of
this structure. We therefore propose a radically compositional approach to
computational neuroscience, drawing on the methods of applied category theory.
We describe how these tools grant us a means to overcome complexity and
improve interpretability, and supply a rigorous common language for scientific
modelling, analogous to the type theories of computer science. As a case
study, we sketch how to translate from compositional narrative concepts to
neural circuits and back again.

##  1 Introduction

How should we conceive of concepts, of cognition, or of circuit computation?
We argue: compositionally. Composition is the tool by which we construct
complex concepts: constantly, informally, and automatically. But it is not
just our concepts that are compositional: it has long been noted that our
cognitive architecture is modular [fodor1983modularity, marcus2018deep], and
much of cognitive neuroscience relies on the well-tested assumption that this
modularity maps onto the structure of the brain. Mathematics itself is
increasingly recognized as compositional [kock2006synthetic, bauer2017hott],
and strong compositionality is increasingly used in software engineering to
improve correctness and code reusability [fischer2017haskell,
haftmann2010higher]. Despite this simplifying power, few models in cognitive
science and machine learning are actually compositional, even when modelling
compositionality itself (_eg._ , [chang2016compositional,
piantadosi2016logical]).

We propose instead taking compositionality seriously, using the mathematics of
composition—category theory—and show how doing so allows us to translate
concepts between contexts and levels: from abstract concepts themselves
[phillips2010categorial, bolt2019interacting] to their possible realization in
a circuit model. This paper is an abridged version of a work in progress,
provisionally to appear in _Compositionality_ , and we defer many formal
details and proofs to that manuscript. In §2 we introduce the background
mathematics. In §LABEL:sec:concepts, we introduce the conceptual setting.
§LABEL:sec:circ shows how to translate concepts to circuits, and
§LABEL:sec:back suggests how to translate back again.

##  2 Category theory and the ‘structure of structure’

A category is a very simple structure, capturing only what is necessary to
enforce compositionality:

######  Definition 1.

A category $\operatorname{\mathcal{C}}$ is a set of objects
$\operatorname{\mathcal{C}}_{0}$ such that for any two objects
$x,y\in\operatorname{\mathcal{C}}_{0}$ there is a set
$\operatorname{\mathcal{C}}(x,y)$ of arrows ${x\to y}$ obeying a composition
rule: for any arrow $f:x\to y$ and any arrow $g:y\to z$, there is a composite
arrow $g\circ f:x\to z$. Every object $x\in\operatorname{\mathcal{C}}_{0}$ has
an identity arrow,
$\operatorname{\mathsf{id}}_{x}\in\operatorname{\mathcal{C}}(x,x)$.

Being so general, almost all concepts can be formalized categorically. For
example, there is a category whose objects are parts of speech and whose
arrows are grammatical relations; and a category of vector spaces and linear
maps. Category theory can also be applied to itself: there is a category
$\mathbf{Cat}$ of categories, whose arrows are called functors:


# Towards the standard Majorana representations of 3-transposition groups

Albert Gevorgyan111Research supported by Imperial College London President’s
PhD Scholarships<EMAIL_ADDRESS>Department of Mathematics,
Imperial College, 180 Queen’s Gt., London, SW7 2AZ, UK

###### Abstract

In this paper, we discuss $3$-transposition groups. In particular, we find
sizes of maximal symmetric subgroups of the groups, which are in Fischer list.
In addition, we build faithful representations of symmetric groups in
orthogonal, symplectic, and unitary spaces, which fully verify the main
result. Thus, we find candidate groups, for which one may be able to build
standard Majorana representations.

###### keywords:

3-transposition group, Fischer group, symmetric subgroup, Monster group,
Majorana representation

###### MSC:

[2010] 05E15 , 20E07

††journal: Journal of LaTeX Templates

##  1 Introduction

Assume that $G$ is a group generated by a set of its involutions $T$, which is
a union of conjugacy classes of $T$. In addition, let $W$ be a linear
representation of $G$, which is also equipped with a positive definite form
and an algebra product.

######  Definition 1.

Let the quintet

$(G,T,W,\varphi,\psi)$

be such that $\varphi:G\rightarrow Aut(W)$ is faithful, $\psi:T\rightarrow W$
is injective with $(t^{\psi})^{g^{\psi}}=(t^{g})^{\psi}$ for all $g\in G,t\in
T$. If the quintet satisfies the axioms (M1)-(M8) described in [6], then it is
called a Majorana representation of $G$.

From the definition one may deduce that if $T^{\prime}\subseteq T$ and $H$ is
the subgroup of $G$ generated by the involutions in $T^{\prime}$, then
$(H,T^{\prime},W,\varphi|_{H},\psi|_{T^{\prime}})$ is a Majorana
representation of $H$.

In [5] the author shows that if the group $G$ admits a Majorana
representation, then the order of the product of any pair of involutions from
$T$ is at most $6$. Later in [6] the authors build all the existing Majorana
representations of the group $S_{4}$. Moreover, in [7] and [8] authors build
Majorana representations of the groups $A_{5}$, $A_{6}$, and $A_{7}$, where
there the set $T$ of the involutions is the set of bitranspositions (the set
of elements of $A_{n}$ of cycle type $2^{2}$).

In [4], the authors discuss whether the group $A_{13}$ admits a standard
Majorana representation. The authors assume that such a representation $V$
exists and under that assumption show that the form defined on $V$ is not
positive definite. Thus, they deduce that if $A_{n}$ admits a standard
Majorana representation, then $n\leq 12$.

######  Definition 2.

([1]) Let $n\geq 3$ be an integer. We call the pair $(G,D)$, where $G$ is a
group and $D$ is a set of involutions of $G$ stable under conjugation, an
$n$-transposition group, if the set $D$ generates $G$ and $o(d_{1}d_{2})\leq
n$ for all $d_{1},d_{2}\in D$.

Sometimes, when it is obvious from the context to which involutions $D$ we
refer, by abuse of terminology, we say that ”$G$ is an $n$-transpostion
group”. In [5] it is shown that the Monster group $M$ is generated by a
conjugacy class of $2A$-involutions. It is also known that the order of the
product of any two $2A$-involutions is at most $6$. Therefore, we obtain that
$(M,2A)$ is a $6$-transposition group. Furthermore, there exists a Majorana
representation $(M,2A,V_{M},\varphi,\psi)$ of the Monster group $M$.

The main objects of interest of this paper are the $3$-transposition groups.
Our main goal is to find groups $G$, which are candidates to admit Majorana
representations.

######  Theorem 1.

Suppose $(G,D)$ is a finite $3$-transposition group. Let $S=\\{p\cdot q|p\neq
q\in D,[p,q]=1\\}$ and $H$ be the subgroup generated by the elements in $S$.
Then, $(H,S)$ is a $6$-transposition group.

This theorem is proven in [9] and is an important starting point for this
paper. In particular, it is known that some $3$-transposition groups are
embedded into the Monster, so that the images of the elements of $S$ are
$2A$-involutions. In addition, if such an embedding does not exist, there may
still exist a Majorana representation for these groups.

######  Theorem 2.

(Fischer, [1, 2, 3]) Assume that $(G,D)$ is a $3$-transposition group so that
$Z(G)$ is trivial and $[G,G]$ is simple. Then one of the following holds:

  * 1.

$G\cong S_{n}$, $D$ is the set of all transpositions;

  * 2.

$G\cong Sp_{n}(2)$, $D$ is the set of transvections $t(v)$, where $v\neq 0$;

  * 3.

$G\cong U_{n}(2)$, $D$ is the set of transvections $t(v)$, where $v\neq 0$ is
singular;

  * 4.

$G\cong O_{n}^{\epsilon}(2)$, $D$ is the set of transvections $t(v)$, where
$v$ is non-singular;

  * 5.

$G\cong PO_{n}^{\mu,\pi}(3)$, $D$ is the conjugacy class of reflections
$t(v)$, where $t(v)=\pi$;

  * 6.

$G$ is of type $M(22),M(23)$ or $M(24)$ and $D$ is a uniquely determined class
of transpositions of $G$;

In this paper for each of above groups $(G,D)$ we find the maximal number $n$
, so that there exists a monomorphism $\varphi:S_{n}\rightarrow G$ with
$\varphi(ij)\in D$, for all transpositions $(ij)$. If we have embedding
$\varphi:S_{n}\rightarrow G$, then we may restrict it to
$\varphi^{\prime}:A_{n}\rightarrow G$, and images of bitranspositions are the
products of commuting transpositions of $D$, which form a set of
$6$-transpositions by the theorem 1. Thus, if $G$ admits a standard Majorana
representation, then so does $A_{n}$. Therefore, $n\leq 12$. We will show that
the classification of the candidate groups is the following:

######  Theorem 3.

Assume that $G$ is a group from the Fischer’s list such that $\varphi(G)\leq
12$. Then one of the following holds:

  * 1.

$G\cong S_{n}$, with $n\leq 12$;

  * 2.

$G\cong Sp_{n}(2)$, with $n\leq 10$;

  * 3.

$G\cong U_{n}(2)$, with $n\leq 11$;

  * 4.

$G\cong{O_{n}}^{+,+}(3)$, with $n\leq 11$;

  * 5.

$G\cong{O_{n}}^{-,+}(3)$, with $n\leq 12$;

  * 6.

$G\cong{O_{n}}^{\epsilon}(2)$, with $n\leq 11$;

  * 7.

$G$ is one of the Fischer groups $M(22)$, $M(23)$ and $M(24)$.

In this paper, we adopt an elementary combinatorial approach for finding the
maximal size of symmetric subgroups of these groups. We do not consider any
character tables and all the calculations are carried out by hand. In
addition, we build geometrical representations of these groups, which verify
the main result for the classical groups.

##  2 Notation

In this paper we use the following notation for the classical groups, which
agrees with the notation of [1]:

  * 1.

For an even integer $n$ we write $Sp_{n}(2)$ for the isometry group of the
$n$-dimensional symplectic space $V$ over the $2$-element field. This group is
generated by transvenctions $t(v):w\rightarrow w+(v,w)v$, where $v$ is a non-
zero vector.

  * 2.

For an integer $n$ we write $U_{n}(2)$ for the isometry group of the
$n$-dimensional unitary space $V$ over the $4$-element field. This group is
generated by transvenctions $t(v):w\rightarrow w+(v,w)v$, where $v$ is a non-
zero singular vector.

  * 3.

For an integer $n$ we write $O^{\epsilon}_{n}(2)$ ($\epsilon\in\\{+,-\\}$) for
the isometry group of the $n$-dimensional orthogonal space $V$ of type
$\epsilon$ over the $2$-element field. This group is generated by
transvenctions $t(v):w\rightarrow w+(v,w)v$, where $v$ is a non-singular
vector.

  * 4.

For an integer $n$ we write $O^{\mu,\pi}_{n}(3)$ ($\mu,\pi\in\\{+,-\\}$) for
the subgroup of the isometry group of the $n$-dimensional orthogonal space $V$
over the $3$-element field, a Gram matrix of a basis of which has determinant
equal to $\mu$, generated by the reflections $t(v)$ for vectors with
$Q(v)=\pi$.

In each case $t(v)$ and $t(u)$ commute if and only if $(v,u)=0$.

It also known that, when $n$ is odd, then $O_{n}^{+}(2)\cong O_{n}^{-}(2)\cong
Sp_{n-1}(2)$. Furthermore, $O_{n}^{\mu,\pi}(3)\cong O_{n}^{\mu,-\pi}(3)$, if
$n$ is even, and $O_{n}^{\mu,\pi}(3)\cong O_{n}^{-\mu,-\pi}(3)$, if $n$ is
odd.

In what follows, $\varphi(G)$ denotes the maximal number $n$ , so that there
exists a monomorphism $\varphi:S_{n}\rightarrow G$ with $\varphi(ij)\in D$,
for all transpositions $(ij)$.

##  3 Auxiliary results

In this section, we prove several lemmas, which later help us to prove the
results of the main part.

######  Definition 3.

Let $(G,D)$ be a $3$-transposition group. We define a chain of transpositions
as a sequence of pairwise distinct transpositions $g_{1},...,g_{m}$, for which
$[g_{i},g_{j}]=1$ (with $i\neq j$) if and only if $|i-j|\neq 1$. We call $m$
the length of the chain.

Recall that, when $G$ is an orthogonal, symplectic or unitary group, then the
transpositions are of form $t(v)$ for some vectors $v$ of the corresponding
vector space, and the transpositions commute if and only if the corresponding
vectors are orthogonal. Thus, while working with such groups, if the
transpositions $g_{1}=t(v_{1}),...,g_{m}=t(v_{m})$ form a chain, we say that
the vectors $v_{1},...,v_{m}$ form a chain of length $m$. Note that if
$v_{1},...,v_{m}$ form a chain, then no pair of vectors may be collinear.

######  Lemma 1.

(big) Let $(G,D)$ be a $3$-transposition group. Let $m$ be the length of the
maximal chain of transpositions of $G$. Then, $\varphi(G)=m+1$.

###### Proof.

First, let us note that if there exists a monomorphism $\psi:S_{n}\rightarrow
G$, then the transpositions $d_{1}=\psi(1,2)$, $d_{2}=\psi(2,3)$, …,
$d_{n-1}=\psi(n-1,n)$ form a chain. Hence, $m\geq\varphi(G)-1$, so
$\varphi(G)\leq m+1$.

On the other hand, suppose there exist transpositions $d_{1},...,d_{m}$
forming a chain. Then, if we create the non-commuting graph of
$\\{d_{1},...,d_{m}\\}$, it is a Coxeter graph of form $A_{m}$. It is well-
known fact that the corresponding Coxeter group is isomorphic to $S_{m+1}$.
Thus, there exists a homomorphism $\psi:S_{m+1}\rightarrow<d_{1},...,d_{m}>$
with $\psi(i,i+1)=d_{i}$.

Note that $ker(f)\unlhd S_{m+1}$. Moreover, the normal subgroups of $S_{m+1}$
are $\\{1\\}$, $A_{m+1}$ and $S_{m+1}$ and possibly the Klein $4$-group (if
$m+1=4$). In addition, if $ker(f)\neq\\{1\\}$, then the factor group
$S_{m+1}/ker(f)$ does not contain $m$ involutions. Therefore, $ker(f)$ is
trivial and $f$ is an isomorphism. Thus $\varphi(G)\geq m+1$. ∎

In addition, we use the following classical result, which can be found in [1].

######  Lemma 2.

(Witt): Let $V$ be an orthogonal, symplectic or unitary space. Let $U$ and $W$
be subspaces of $V$ and suppose that $\alpha:U\rightarrow W$ is an isometry of
subspaces. Then, $\alpha$ extends to an isometry of $V$.

######  Lemma 3.

(small): Let $V$ be an orthogonal, symplectic or unitary space. Let the
vectors $v_{1},...,v_{n+1}\in V$ form a chain. Then the vectors
$v_{1},...,v_{n}$ are linearly independent.

###### Proof.

Suppose, for a contradiction, there exists a non-trivial linear combination
$c_{1}v_{1}+...+c_{n}v_{n}=0$. Let $m$ be the maximal number with $c_{m}\neq
0$. Then, $c_{1}v_{1}+...+c_{m}v_{m}=0$. Therefore,
$0=(0,v_{m+1})=(c_{1}v_{1}+...+c_{m}v_{m},v_{m+1})=c_{1}(v_{1},v_{m+1})+...+c_{m}(v_{m},v_{m+1})=c_{m}(v_{m},v_{m+1})\neq
0$, which is a contradiction. ∎

######  Lemma 4.

(main): Let $V$ be an orthogonal, symplectic or unitary space. Let
$v_{1},...,v_{m}$ be linearly independent vectors in $V$, which form a chain.
Then, $v_{1},...,v_{m}$ can be continued to a maximal chain of $V$.

###### Proof.

Let $w_{1},...,w_{n}$ be a maximal chain of $V$. If $n=m$, then the statement
is true. Otherwise, $m\leq n-1$, so by the small lemma the vectors
$w_{1},...,w_{m}$ are linearly independent. Furthermore, let
$w_{1}^{\prime}=w_{1}$. Then, we can substitute $w_{2}$ with
$w_{2}^{\prime}=c_{2}w_{2}$ for some constant $c_{2}\neq 0$ so that
$(v_{1},v_{2})=(w_{1}^{\prime},w_{2}^{\prime})$. Similarly, we can substitute
$w_{3},...,w_{m}$ with their scalar multiples so that
$(v_{i},v_{i+1})=(w_{i}^{\prime},w_{i+1}^{\prime})$ for $i=1,...,m-1$. Let
$U=span(v_{1},...,v_{m})$ and $W=span(w_{1},...,w_{m})$. Then,
$v_{1},...,v_{m}$ form a basis for $U$ and $w_{1}^{\prime},...,w_{m}^{\prime}$
form a basis for $W$. Hence, we can define a linear map $L:W\rightarrow U$
with $L(w_{i}^{\prime})=v_{i}$ for $i=1,...,m$. Then, by the choice of
$w^{\prime}_{1},...,w^{\prime}_{m}$, $L$ is an isometry. Therefore, by the
Witt’s lemma, $L$ can be extended to an isometry $T$ of $V$. Denote,
$v_{m+1}=T(w_{m+1})$, …, $v_{n}=T(w_{n})$. From the fact that $T$ is isometry,
one may deduce that $v_{1}=T(w_{1}^{\prime})$, …, $v_{n}=T(w_{n})$ is a chain
as well. ∎

It is well known that if $V$ is a symplectic, unitary or orthogonal space,
$v\in V$ and $c$ is a constant, then $t(v)=t(cv)$. Therefore, if we have a
chain of vectors $v_{1},...,v_{n}$ and multiply some of these vectors by non-
zero constants, then it will also be a chain.

##  4 Main results

Now, after stating and proving important lemmas in the previous sections, we
are ready for proving the main results of this paper. We compute the value of
$\varphi(G)$ for each group in Fischer’s list.

######  Proposition 1.

If $G\cong S_{n}$, then $\varphi(G)=n$.

The statement is elementary.

######  Proposition 2.

If $G\cong Sp_{2n}(2)$ and $n\geq 2$, then $\varphi(G)=2n+2$.

###### Proof.

Assume that $V$ is a symplectic space of dimension $2n$ over the $2$-element
field. Let $v_{1},...,v_{2n}$ be a basis for $V$ with $(v_{i},v_{j})=1$ if
$\\{i,j\\}\in\\{\\{1,2\\},\\{3,4\\},...,$ $\\{2n-1,2n\\}\\}$ and
$(v_{i},v_{j})=0$ otherwise.

Note that the following vectors are linearly independent and form a chain:

$v_{1}+v_{2},v_{2}+v_{3},...,v_{2n-1}+v_{2n},(v_{1}+v_{2}+...+v_{2n-2})+v_{2n}.$

Thus, by the main lemma, this may be continued to a maximal chain. Let
$c_{1}v_{1}+...+c_{2n}v_{2n}$ be the next vector in the chain. Then, it is
orthogonal to each of the vectors

$v_{1}+v_{2},v_{2}+v_{3},...,v_{2n-1}+v_{2n}.$

Therefore, $c_{1}=c_{2}=...=c_{2n}$. Moreover, the vector is non-zero,
therefore $c_{1}=1$. Hence, the next vector in this chain is
$v_{1}+...+v_{2n}$. Note that this chain cannot be continued anymore, as there
are no other non-zero vectors orthogonal to vectors
$v_{1}+v_{2},v_{2}+v_{3},...,v_{2n-1}+v_{2n}$. Hence, the maximal chain has
length $2n+1$, so by the big lemma, $\varphi(G)=(2n+1)+1=2n+2$. ∎

######  Proposition 3.

Let $G\cong U_{n}(2)$. Then, $\varphi(G)=n+1$, when $n\geq 5$ is odd, and
$\varphi(G)=n+2$, when $n\geq 4$ is even.

###### Proof.

Assume that $V$ is a unitary space of dimension $n$ over the $4$-element field
$\mathbb{F}_{4}$. Let $v_{1},...,v_{n}$ be an orthonormal basis. We write
$0,1,\alpha,\overline{\alpha}$ for the elements of $\mathbb{F}_{4}$ and
$\overline{x}$ for the conjugate of an element $x$ in $\mathbb{F}_{4}$.

Now, assume that $n\geq 5$ is odd. Then, the following vectors form a chain
and are linearly independent:

$v_{1}+v_{2},v_{2}+v_{3},...,v_{n-1}+v_{n}.$

According to the main lemma, this chain can be continued to a maximal chain.
Let $c_{1}v_{1}+...c_{n}v_{n}$ be the next vector in this chain. Then, it is
orthogonal to the vectors $v_{1}+v_{2},...,v_{n-2}+v_{n-1}$. Therefore,
$c_{1}=c_{2}=...=c_{n-1}$. Moreover, this is a singular vector, so
$0=(c_{1}v_{1}+...+c_{n}v_{n},c_{1}v_{1}+...+c_{n}v_{n})=(n-1)c_{1}\overline{c_{1}}+c_{n}\overline{c_{n}}=c_{n}\overline{c_{n}}$,
so $c_{n}=0$. Therefore, up to a constant multiple, there is a unique vector
$v_{1}+...+v_{n-1}$, which continues the chain. Note that
$(v_{1}+v_{2})+(v_{3}+v_{4})+...+(v_{n-2}+v_{n-1})+(v_{1}+...+v_{n-1})=0$, so
the vectors in this chain are linearly dependent, therefore by the small
lemma, the chain cannot be continued any further. Thus, the maximal chain has
length $n$ and $\varphi(G)=n+1$.

Now, assume that $n\geq 4$ is even. Then, the following vectors are linearly
independent and form a chain:

$v_{1}+v_{2},v_{2}+v_{3},...,v_{n-1}+v_{n},v_{1}+v_{2}+...+v_{n-1}+\alpha
v_{n}.$

According to the main lemma, this chain can be continued to a maximal chain.
Let $c_{1}v_{1}+...+c_{n}v_{n}$ be the next vector in this chain. Then, it is
orthogonal to the vectors $v_{1}+v_{2},...,v_{n-1}+v_{n}$. Hence,
$c_{1}=c_{2}=...=c_{n}$. Therefore, up to a constant multiple, there is a
unique vector $v_{1}+...+v_{n}$, which continues the chain. Note that
$(v_{1}+v_{2})+(v_{3}+v_{4})+...+(v_{n-2}+v_{n-1})+(v_{1}+...+v_{n})=0$, so
the vectors in this chain are linearly dependent. Thus, by the small lemma,
the chain cannot be continued any further. Thus, the maximal chain has length
$n+1$ and $\varphi(G)=(n+1)+1=n+2$. ∎

######  Proposition 4.

Let $G\cong PO_{n}^{+,+}(3)$ and $n\geq 3$. Then, $\varphi(G)=n$, $n\equiv 1$
(mod $3$), and $\varphi(G)=n+1$ otherwise.

###### Proof.

Assume that $(V,Q)$ is an $n$-dimensional orthogonal space over the
$3$-element field. Let $v_{1},...,v_{n}$ be a basis for $V$ with
$(v_{i},v_{j})=0$ if $i\neq j$ and $(v_{i},v_{i})=1$. Note that
$Q(v)=2(v,v)=-(v,v)$.

Then, the following vectors are linearly independent and form a chain:

$v_{1}-v_{2},v_{2}-v_{3},...,v_{n-1}-v_{n}.$

According to the main lemma, this chain can be continued to a maximal chain.
Let $v=c_{1}v_{1}+...+c_{n}v_{n}$ be the next vector in the chain. Then, it is
orthogonal to the vectors $v_{1}-v_{2},v_{2}-v_{3},...,v_{n-2}-v_{n-1}$, so
$c_{1}=c_{2}=...=c_{n-1}$. Note that if $c_{1}=0$, then
$Q(c_{1}v_{1}+...+c_{n}v_{n})=Q(c_{n}v_{n})=c_{n}^{2}Q(v_{n})=-c_{n}^{2}\neq
1$. So, $c_{1}\neq 0$. Note that if $c_{1}=-1$, then we can multiply $v$ by
$-1$. Then, $v=(v_{1}+...+v_{n-1})+c_{n}v_{n}$.

If $n\equiv 0$ (mod $3$), then $1=Q(v)=-(v,v)=-(-1+c_{n}^{2})=1-c_{n}^{2}$,
thus $c_{n}^{2}=0$, which implies $c_{n}=0$. Thus, $v=v_{1}+...+v_{n-1}$. Note
that $v$ is the only vector (up to a constant multiple) with $Q(v)=1$, which
is orthogonal to $v_{1}-v_{2},v_{2}-v_{3},...,v_{n-2}-v_{n-1}$. Thus, this
chain cannot be continued anymore. So, the maximal chain has length $n$ and
$\varphi(G)=n+1$.

If $n\equiv 1$ (mod $3$), then $1=Q(v)=-(v,v)=-(0+c_{n}^{2})=-c_{n}^{2}$,
which is impossible. So, the chain $v_{1}-v_{2},...,v_{n-1}-v_{n}$ is a
maximal chain, and $\varphi(G)=(n-1)+1=n$.

If $n\equiv 2$ (mod $3$), then $1=Q(v)=-(v,v)=-(-1+c_{n}^{2})=1-c_{n}^{2}$, so
$c_{n}^{2}=0$, which implies $c_{n}\in\\{-1,1\\}$. But $v$ is not orthogonal
to $v_{n-1}-v_{n}$, so $c_{n}=-1$. Note that,
$(v_{1}+...+v_{n-1}-v_{n})-(v_{1}-v_{2})+(v_{2}-v_{3})-(v_{4}-v_{5})+(v_{5}-v_{6})-...-(v_{n-4}-v_{n-3})+(v_{n-3}-v_{n-2})-(v_{n-1}-v_{n})=0$.
So, the system is linearly dependent, and by the small lemma this chain cannot
be continued. Therefore, the maximal chain has length $n$ and
$\varphi(G)=n+1$. ∎

######  Proposition 5.

Let $G\cong PO_{n}^{-,+}(3)$ and $n\geq 4$. Then, $\varphi(G)=n$, if $n\equiv
0$ (mod $3$), $\varphi(G)=n+1$, if $n\equiv 2$ (mod $3$) and $\varphi(G)=n+2$,
if $n\equiv 1$ (mod $3$).

###### Proof.

Assume that $(V,Q)$ is an $n$-dimensional orthogonal space over the
$3$-element field. Let $v_{1},...,v_{n}$ be a basis for $V$ with
$(v_{i},v_{j})=0$ if $i\neq j$, $(v_{i},v_{i})=1$, if $i<n$ and
$(v_{n},v_{n})=-1$. Note that $Q(v)=2(v,v)=-(v,v)$.

Now, assume that $n\equiv 1$ (mod $3$) with $n\geq 4$. Then, the following
vectors form a chain:

$\displaystyle
v_{1}-v_{2},v_{2}-v_{3},...,v_{n-2}-v_{n-1},(v_{1}+...+v_{n-2})-v_{n-1}+v_{n},$
$\displaystyle(v_{1}+...+v_{n-2}+v_{n-1})-v_{n},v_{1}+...+v_{n}.$

Moreover, as a consequence of the small lemma, there cannot be a chain of
length more than $n+1$. Thus, the longest chain has length $n+1$ and
$\varphi(G)=(n+1)+1=n+2$.

Now, assume that $n\equiv 2$ (mod $3$) with $n\geq 5$. Then, the following
vectors are linearly independent and form a chain:

$v_{1}-v_{2},v_{2}-v_{3},...,v_{n-2}-v_{n-1},(v_{1}+...+v_{n-2})+v_{n}.$

Therefore, this chain can be continued to a maximal chain. Let
$v=c_{1}v_{1}+...+c_{n}v_{n}$ be the next vector. Then, it is orthogonal to
vectors $v_{1}-v_{2},v_{2}-v_{3},...,v_{n-2}-v_{n-1}$. Therefore,
$c_{1}=...=c_{n-1}$. Note that if $c_{1}\neq 0$, then
$1=Q(v)=(n-1)c_{1}^{2}+c_{n}^{2}=(n-1)+c_{n}^{2}=1+c_{n}^{2}$, so $c_{n}=0$.
But then, $(v,(v_{1}+...+v_{n-2})+v_{n})=n-2=0$. So, $c_{1}=0$. Then,
$v=v_{n}$ (or $-v_{n}$, these differ by a multiple of $-1$). Note that this
chain cannot be continued, because
$((v_{1}+...+v_{n-2})+v_{n})-(v_{1}-v_{2})+(v_{2}-v_{3})-(v_{4}-v_{5})+(v_{5}-v_{6})-...-(v_{n-4}-v_{n-3})+(v_{n-3}-v_{n-2})-v_{n}=0$,
so the vectors in this chain are linearly dependent. So, the maximal chain has
length $n$ and $\varphi(G)=n+1$.

Finally, assume that $n\equiv 0$ (mod $3$) and $n\geq 6$. Then, the following
vectors are linearly independent and form a chain:

$v_{1}-v_{2},v_{2}-v_{3},...,v_{n-2}-v_{n-1}.$

By the main lemma, this can be continued to a maximal chain. Let
$c=c_{1}v_{1}+...+c_{n}v_{n}$ be the next vector. Then, $c_{1}=...=c_{n-2}$.
Then $1=Q(v)=-(v,v)=-(1+c_{n-1}^{2}-c_{n}^{2})$. The only possibility is
$c_{n}=0$ and $c_{n-1}\neq 0$. But then $(v,v_{n-2}-v_{n-1})\neq 0$, so
$c_{n-1}=-1$. So, $v=(v_{1}+...+v_{n-2})-v_{n-1}$. Note that this chain cannot
be continued, because
$((v_{1}+...+v_{n-2})-v_{n-1})-(v_{1}-v_{2})+(v_{2}-v_{3})-(v_{4}-v_{5})+(v_{5}-v_{6})-...+(v_{n-4}-v_{n-3})-(v_{n-2}-v_{n-1})=0$,
is linearly dependent. Thus, the maximal chain has length $n-1$ and
$\varphi(G)=(n-1)+1=n$. ∎

In [1] it is proven that if $n$ is odd, then $PO_{n}^{+,-}(3)\cong
PO_{n}^{-,+}(3)$ and $PO_{n}^{-,-}(3)\cong PO_{n}^{+,+}(3)$. Moreover, if $n$
is even, then $PO_{n}^{+,-}(3)\cong PO_{n}^{+,+}(3)$ and $PO_{n}^{-,-}(3)\cong
PO_{n}^{-,+}(3)$. Therefore, the previous propositions can be used to find the
results for the groups $PO_{n}^{\mu,-}(3)$ as well. Moreover, if $n$ is odd,
then $O_{n}^{+}(2)\cong O_{n}^{-}(2)\cong Sp_{n-1}(2)$ as a $3$-transposition
group.

######  Proposition 6.

Let $G\cong O_{n}^{+}(2)$, $n\geq 6$ be even. Then, $\varphi(G)=n$, if
$n\equiv 2$ (mod $8$), $\varphi(G)=n+2$, if $n\equiv 6$ (mod $8$) and
$\varphi(G)=n+1$, if $4|n$ and $n\geq 8$.

###### Proof.

Assume that $V$ is an $2n$-dimensional orthogonal space of $+$ type. Let
$v_{1},...,v_{n}$ be a basis with $Q(v_{1})=...=Q(v_{n-1})=1$ and $Q(v_{n})=1$
if and only if $4|n$. Also,
$(v_{1},v_{2})=(v_{3},v_{4})=...=(v_{n-1},v_{n})=1$ and the other products are
$0$.

First, assume $8|n$. Then the following vectors are linearly independent and
form a chain:

$\displaystyle
v_{1}+v_{2},v_{2}+v_{3}+v_{n-1},v_{3}+v_{4},v_{4}+v_{5}+v_{n-1},...,v_{n-3}+v_{n-2},$
$\displaystyle(v_{1}+...+v_{n-4})+v_{n-2},v_{1}+v_{2}+v_{5}+v_{6}+...+v_{n-3}+v_{n-2}+v_{n},v_{n-1}.$

By the main lemma, this chain can be continued to a maximal chain. Note that
if the set $w_{1},...,w_{n}$ is a basis for $V$, then the system of equations
$(w,w_{1})=...=(w,w_{n-1})=0$, $(w,w_{n})=1$ has a unique solution. Note that
$w_{1}+w_{3}+...+w_{n-1}$ is a solution for this system. However,
$Q(w_{1}+w_{3}...+w_{n-1})=\frac{n}{2}=0$ is singular. Thus, the maximal chain
has length $n$ and $\varphi(G)=n+1$.

Now, assume $n\equiv 2$ (mod $8$). Then the following vectors are linearly
independent and form a chain:

$\displaystyle
v_{1}+v_{2},v_{2}+v_{3}+v_{n-1},v_{3}+v_{4},v_{4}+v_{5}+v_{n-1},...,v_{n-3}+v_{n-2},$
$\displaystyle(v_{1}+...+v_{n-4})+v_{n-2}+v_{n-1},v_{1}+v_{2}+...+v_{n-1}.$

By the main equation this chain can be continued to a maximal chain. Let
$v=c_{1}v_{1}+...+c_{n}v_{n}$ be the next vector in the chain. Then, because
of conditions of orthogonality, $c_{1}=c_{2}$, $c_{3}=c_{4}$, …,
$c_{n-3}=c_{n-2}$. In addition,
$c_{1}+c_{3}+c_{n}=c_{3}+c_{5}+c_{n}=...=c_{n-5}+c_{n-3}+c_{n}=0$. In
addition, $c_{n-3}+c_{n}=0$ and $c_{n}=1$. So,
$v=(v_{3}+v_{4})+(v_{7}+v_{8})+...+(v_{n-3}+v_{n-2})+(cv_{n-1}+v_{n})$, which
is singular. Thus, this is a maximal chain and $\varphi(G)=(n-1)+1=n$.

First, assume $n\equiv 4$ (mod $8$). Then the following vectors are linearly
independent and form a chain:

$\displaystyle v_{1}+v_{2},v_{2}+v_{3}+v_{n-1},v_{3}+v_{4},$ $\displaystyle
v_{4}+v_{5}+v_{n-1},...,v_{n-3}+v_{n-2},(v_{1}+...+v_{n-4})+v_{n-2},$
$\displaystyle
v_{1}+v_{2}+...+v_{n-2},(v_{3}+v_{4})+(v_{7}+v_{8})+...+(v_{n-5}+v_{n-4})+v_{n}.$

By the main lemma, this chain can be continued to a maximal chain. Note that
if the set $w_{1},...,w_{n}$ is a basis for $V$, then the system of equations
$(w,w_{1})=...=(w,w_{n-1})=0$, $(w,w_{n})=1$ has a unique solution. Note that
$w_{1}+w_{3}+...+w_{n-1}$ is a solution for this system. However,
$Q(w_{1}+w_{3}...+w_{n-1})=\frac{n}{2}=0$ is singular. Thus, the maximal chain
has length $n$ and $\varphi(G)=n+1$.

First, assume $n\equiv 6$ (mod $8$). Then the following vectors are linearly
independent and form a chain:

$\displaystyle
v_{1}+v_{2},v_{2}+v_{3}+v_{n-1},v_{3}+v_{4},v_{4}+v_{5}+v_{n-1},...,v_{n-3}+v_{n-2},$
$\displaystyle(v_{1}+...+v_{n-4})+v_{n-2}+v_{n-1},$ $\displaystyle
v_{1}+v_{2}+...+v_{n-1},(v_{3}+v_{4})+(v_{7}+v_{8})+...+(v_{n-5}+v_{n-4})+v_{n},v_{n-1}.$

Note that this is a maximal chain, because by the small lemma, the chain has
length at most $n+1$. Therefore, $\varphi(G)=(n+1)+1=n+2$. ∎

######  Proposition 7.

Let $G\cong O_{n}^{-}(2)$, $n\geq 4$ be even. Then, $\varphi(G)=n$, if
$n\equiv 6$ (mod $8$), $\varphi(G)=n+2$, if $n\equiv 2$ (mod $8$),
$\varphi(G)=n+1$, if $4|n$.

The proof of this proposition is analogical to the proof of the previous
proposition.

######  Proposition 8.

For the small groups we have the following results:

  * 1.

$\varphi(Sp_{2}(2))=3$.

  * 2.

$\varphi(U_{1}(2))=1$.

  * 3.

$\varphi(U_{2}(2))=3$.

  * 4.

$\varphi(U_{3}(2))=3$.

  * 5.

$\varphi(PO^{+,+}_{1}(3))=1$.

  * 6.

$\varphi(PO^{+,+}_{2}(3))=2$.

  * 7.

$\varphi(PO^{-,+}_{1}(3))=2$.

  * 8.

$\varphi(PO^{-,+}_{2}(3))=2$.

  * 9.

$\varphi(PO^{-,+}_{3}(3))=2$.

  * 10.

$\varphi(O^{+}_{2}(2))=2$.

  * 11.

$\varphi(O^{+}_{4}(2))=3$.

  * 12.

$\varphi(O^{-}_{2}(2))=3$.

###### Proof.

These results can be verified elementary using the technique of the proof of
the above propositions. ∎

######  Proposition 9.

Let $G$ be of type $M(22)$. Then $\varphi(G)=10$.

###### Proof.

Let $x_{1},...,x_{n}$ be a chain of transpositions of $G$. Then,
$x_{1},...,x_{n-2}$ is a chain of transpositions for
$C_{x_{n}}(G)/<x_{n}>\cong U_{6}(2)$. Therefore, by the previous propositions,
$n-2\leq 7$ and $n\leq 9$.

Now, let us construct a chain of length $9$. Let us pick a pair of non-
commuting transpositions $x_{1},x_{2}$. Then, $C_{x_{1},x_{2}}(G)\cong
PO_{6}^{+,+}(3)$. Therefore, there exists a chain of involutions
$x_{4},...,x_{9}$ commuting with $x_{1}$ and $x_{2}$. Now, let
$f:C_{x_{1}}(G)/<x_{1}>\rightarrow U_{6}(2)$ be an isomorphism. Then,
$f(x_{i})=t(v_{i})$, where $v_{i}$ are non-zero singular vectors. Let
$v_{3}=v_{5}+v_{7}+v_{9}$. Then, $v_{3}\neq 0$ is singular. Denote,
$x_{3}=f^{-1}(t(v_{i}))$. Note that $x_{3},...,x_{9}$ is a chain in
$C_{x_{1}}(G)$, but not in $C_{x_{1},x_{2}}(G)$, because the maximal chain
there has length $6$. So, $x_{2}$ and $x_{3}$ do not commute. Hence
$x_{1},...,x_{9}$ is a chain and it is maximal. Therefore,
$\varphi(G)=9+1=10$. ∎

######  Proposition 10.

Let $G$ be of type $M(23)$. Then, $\varphi(G)=12$.

###### Proof.

Let $x_{1},...,x_{n}$ be a chain of transpositions of $G$. Then,
$x_{1},...,x_{n-2}$ is a chain of transpositions for $C_{x_{n}}(G)/<x_{n}>$,
which is of type $M(22)$. Therefore, by the previous proposition, $n-2\leq 9$
and $n\leq 11$. Furthermore, let us prove the following lemma.

######  Lemma 5.

Let $H$ be a group of $M(22)$ type and $g_{1},...,g_{8}$ be a chain of
transpositions. Then, there exists a unique transposition $g_{9}$ so that
$g_{1},...,g_{9}$ is a chain.

###### Proof.

First of all, as we can see from the proof of the previous part if such an
element $g_{9}$ exists, then $g_{3},...,g_{9}\in C_{H}(g_{1})$, so $g_{9}$ is
defined uniquely. Therefore, it suffices to prove that the number of the
chains of length $8$ and the number of chains of length $9$ are equal. Let
$K\cong PO_{6}^{+,+}(3)$ and for $1\leq n\leq 6$ and let $v_{1},...,v_{n-1}$
form a chain. Let $k_{n}$ be the number of vectors $v_{n}$ so that
$v_{1},...,v_{n}$ is a linearly independent chain. By the main lemma, this
number is independent of the choice of $v_{1},...,v_{n}$.

Now, let us compute the number of $9$-chains: there are $3510$ choices for
$g_{1}$ and then there are $2816$ choices for $g_{2}$. Then,
$C_{g_{1},g_{2}}\cong PO_{6}^{+,+}(3)$. Thus, there are $k_{1}$ choices for
$g_{9}$, $k_{2}$ choices for $g_{8}$, …, $k_{6}$ choices for $g_{4}$. Finally,
from the proof of the previous part, $g_{1},g_{2},g_{4},g_{5},...,g_{9}$
define $g_{3}$ in a unique way. Thus, the number of the $9$-chains is
$3510\cdot 2816\cdot k_{1}\cdot...\cdot k_{6}$. Now, let us compute the number
of $8$-chains. Again, there are $3510\cdot 2816$ choices for the ordered pair
$(g_{1},g_{2})$, $k_{1}$ choices for $g_{8}$, …, $k_{5}$ choices for $g_{4}$.
Note that $g_{4},...,g_{8}$ is a chain in $C_{g_{1}}(H)$. Let
$f:C_{g_{1}}(H)/<g_{1}>\rightarrow SU_{6}(2)$ be an isomorphism and
$f(g_{i})=t(v_{i})$. Then, the system of equations of $(v_{3},v_{4})=1$,
$(v_{3},v_{5})=...=(v_{3},v_{8})=0$ has $2$ solutions (by Witt’s lemma, it is
sufficient to verify this argument just for one example - pick
$v_{4}=(110000)$, $v_{5}=(011000)$, …, $v_{8}=(000011)$ with respect to the
unitary basis, then there are two such vectors
$(\overline{\alpha}\alpha\alpha\alpha\alpha\alpha)$ and
$(\alpha\overline{\alpha}\overline{\alpha}\overline{\alpha}\overline{\alpha}\overline{\alpha})$).
Hence, there are $2$ such involutions $g_{3}=f^{-1}(t(v_{3}))$, $k_{6}$ of
which are in $C_{g_{1},g_{2}}(H)$. Therefore, the number of $8$-chains is
$3510\cdot 2816\cdot k_{1}\cdot...\cdot k_{5}\cdot(2-k_{6})$. Thus, it is
sufficient to verify that $3510\cdot 2816\cdot k_{1}\cdot...\cdot
k_{6}=3510\cdot 2816\cdot k_{1}\cdot...\cdot k_{5}\cdot(2-k_{6})$ or
equivalently $k_{6}=1$. $k_{6}$ does not depend on the choice
$v_{1},...,v_{5}$ so pick $v_{1}=(120000)$, $v_{2}=(012000)$, …,
$v_{5}=(000012)$ with respect to the orthogonal basis. Then, if
$v_{6}=(c_{1}c_{2}c_{3}c_{4}c_{5}c_{6})$, then $c_{1}=c_{2}=...=c_{5}$.
Moreover, $Q(v_{6})=1$, so $c_{6}=0$. Hence, either $v_{6}=(111110)$ or
$v_{6}=(222220)$, so $g_{6}=t(111110)=t(222220)$ is unique! ∎

Now, let us construct a chain of length $11$. Let us pick a pair of non-
commuting transpositions $x_{1},x_{2}$. Then, $C_{x_{1},x_{2}}(G)\cong
PO_{7}^{+,-}(3)$. Therefore, there exists a chain of involutions
$x_{4},...,x_{11}$ commuting $x_{1}$ and $x_{2}$. By the lemma, we can
continue the chain $x_{11},...,x_{4}$ of $C_{x_{1}}(G)/<x_{1}>$ with $x_{3}$.
Note that $x_{3},...,x_{11}$ is a chain in $C_{x_{1}}(G)$, but not in
$C_{x_{1},x_{2}}(G)$, because the maximal chain there has length $8$. So,
$x_{2}$ and $x_{3}$ do not commute. Hence $x_{1},...,x_{11}$ is a chain and it
is maximal. Therefore, $\varphi(G)=11+1=12$. ∎

######  Proposition 11.

Let $G$ be of type $M(24)$. Then, $\varphi(G)=12$.

###### Proof.

Firstly, note that $G$ has a subgroup of type $M(23)$, so $\varphi(G)\geq 12$.
Furthermore, if $g_{1},...,g_{n}$ is a chain of transpositions, then
$g_{4},...,g_{n}\in C_{g_{1},g_{2}}(G)$, which is isomorphic to a triple cover
of $PO^{+,+}_{8}(3)$. By the previous propositions, the maximal chain chain
there has length $8$. So, $n-3\leq 8$ and $n\leq 11$. So, $\varphi(G)\leq
11+1=12$. Thus, $\varphi(G)=12$, as desired. ∎

##  5 Geometrical interpretation of the main results

In this section, we discuss the inverse of the problem in the previous section
for the classical groups. We build representations of symmetric groups $S_{n}$
in the spaces with symplectic, unitary, or orthogonal forms. From the previous
part, one may easily deduce, that if $G$ is a symplectic, unitary or
orthogonal group of dimension $n$, then $n\leq\varphi(G)\leq n+2$. Therefore,
we anticipate that $S_{n}$ should act on a space of dimension $m$, with
$n-2\leq m\leq n$. For that purpose, we build a symplectic, unitary, or
orthogonal space $V$ with basis $v_{1},...,v_{n}$ and allow $S_{n}$ to permute
the basis. Then, we take the subspace, which is orthogonal to some vector,
and, if possible, factorize it over some vector. If needed, for fixing the
sign of an orthogonal space, we take a direct sum with a one-dimensional
space.

In this section we look for a representation of $S_{n}$ with $n\geq 5$.
Actually, in this case, the action of $S_{n}$ on the corresponding space is
faithful, because the kernel of the action is a normal subgroup of $S_{n}$,
that is $\\{1\\},A_{n}$ or $S_{n}$. And the image of the representation has
more than two elements, so the kernel is the trivial subgroup.

###  5.1 Symplectic spaces over the 2-element field

Let $n$ be an even integer. Suppose that $W$ is an $n$-dimensional symplectic
space over the $2$-element field. Let $w_{1},...,w_{n}$ be a basis for $W$
with $(w_{i},w_{j})=1$, if $i\neq j$. Let $W^{\prime}=\\{x\in
W:(x,w_{1}+...+w_{n})=0\\}$. Then, $W^{\prime}$ is a degenerate space :
$w_{1}+...+w_{n}\in W^{{}^{\prime}}$. Thus, if we factor $W^{{}^{\prime}}$ by
the subspace generated by the vector $w_{1}+...+w_{n}$ we obtain the space
$V=W^{{}^{\prime}}/<w_{1}+...+w_{n}>$ which is a symplectic space of dimension
$n-2$. Let $\tau\in S_{n}$. Then, let $\psi(\tau):W\rightarrow W$ be a linear
operator for which $\psi(\tau)(w_{i})=w_{\tau(i)}$. Furthermore, one may
restrict $\psi(\tau)$ on $V$. Note that $\psi(\tau)$ preserves the symplectic
form on $V$, so $\psi$ is the natural embedding of $S_{n}$ into $Sp_{n-2}(2)$.
Finally, $\psi(ij)(w_{i})=w_{j}$, $\psi(ij)(w_{j})=w_{i}$, and
$\psi(ij)(w_{k})=w_{k}$ if $k\neq i,j$ in $W$. Note that the transvention
$t(w_{i}+w_{j})$ defined on $W$ acts in the same way on the basis
$(v_{1},...,v_{n})$. On the induced space $V$, $t(w_{i}+w_{j})$ is also a
transvection, as $w_{i}+w_{j}\in W^{\prime}$. Thus, $\psi(ij)=t(w_{i}+w_{j})$
on $V$. Therefore, we verified that the images of transpositions under $\tau$
are transvenctions of $V$.

###  5.2 Unitary spaces over the 4-element field

Let $n$ be an even integer. Suppose that $W$ is an $n$-dimensional unitary
space over the $4$-element field. Let $w_{1},...,w_{n}$ be a basis for $W$
with $(w_{i},w_{i})=1$ and $(w_{i},w_{j})=0$, if $i\neq j$. Let
$W^{\prime}=\\{x\in W:(x,w_{1}+...+w_{n})=0\\}$. Then, $W^{\prime}$ is a
degenerate space : $w_{1}+...+w_{n}\in W^{{}^{\prime}}$. Thus, if we factor
$W^{{}^{\prime}}$ by the subspace generated by the vector $w_{1}+...+w_{n}$ we
obtain the space $V=W^{{}^{\prime}}/<w_{1}+...+w_{n}>$ which is a unitary
space of dimension $n-2$. Let $\tau\in S_{n}$. Then, let
$\psi(\tau):W\rightarrow W$ be a linear operator for which
$\psi(\tau)(w_{i})=w_{\tau(i)}$. Furthermore, one may restrict $\psi(\tau)$ on
$V$. Note that $\psi(\tau)$ preserves the unitary form on $V$, so $\psi$ is
the natural embedding of $S_{n}$ into $U_{n-2}(2)$. Finally,
$\psi(ij)(w_{i})=w_{j}$, $\psi(ij)(w_{j})=w_{i}$, and $\psi(ij)(w_{k})=w_{k}$
if $k\neq i,j$ in $W$. Note that the transvention $t(w_{i}+w_{j})$ defined on
$W$ acts in the same way on the basis $(w_{1},...,w_{n})$. On the induced
space $V$, $t(w_{i}+w_{j})$ is also a transvection, as $w_{i}+w_{j}\in
W^{\prime}$. Thus, $\psi(ij)=t(w_{i}+w_{j})$ on $V$ and $w_{i}+w_{j}$ is
singular. Therefore, we verified that the images of transpositions under
$\tau$ are transvenctions of $V$.

###  5.3 Orthogonal spaces over the $3$-element field

Let $n$ be an integer congruent to $0$ or $1$ modulo $3$. Suppose that $W$ is
an $(n+1)$-dimensional orthogonal space over the $3$-element field of type
”-”. Let $w_{1},...,w_{n},w_{n+1}$ be a basis for $W$ with $Q(w_{i})=-1$,
$(w_{i},w_{i})=1$, for $i=\overline{1,...,n}$, $Q(w_{n+1})=1$,
$(w_{n+1},w_{n+1})=-1$ and $(w_{i},w_{j})=0$, if $i\neq j$. Let
$v=w_{1}+...+w_{n}$ if $n\equiv 0$ (mod $3$) and $v=w_{1}+...w_{n+1}$ if
$n\equiv 1$ (mod $3$). Let $W^{\prime}=\\{x\in W:(x,v)=0\\}$. Then,
$W^{\prime}$ is a degenerate space : $v\in W^{{}^{\prime}}$. Thus, if we
factor $W^{{}^{\prime}}$ by the subspace generated by the vector $v$, we
obtain the space $V=W^{{}^{\prime}}/<v>$ which is a unitary space of dimension
$n-1$. Let $\tau\in S_{n}$. Then, let $\psi(\tau):W\rightarrow W$ be a linear
operator for which $\psi(\tau)(w_{i})=w_{\tau(i)}$. Furthermore, one may
restrict $\psi(\tau)$ on $V$. Note that $\psi(\tau)$ preserves the orthogonal
form on $V$.

If, $n\equiv 1$ (mod $3$), then the images of the vectors
$w_{1}-w_{2},w_{1}-w_{3},...,w_{1}-w_{n}$ form a basis for $V$. Let us
construct the Gram matrix: note that if $i\neq j$, then
$(w_{1}-w_{i},w_{1}-w_{j})=-1$ and $(w_{1}-w_{i},w_{1}-w_{j})=1$. So, the Gram
matrix is of form

$A=\begin{pmatrix}-1&1&\cdots&1\\\ 1&-1&\cdots&1\\\
\vdots&\vdots&\ddots&\vdots\\\ 1&1&\cdots&-1\end{pmatrix}$

Note that, $(A-I)^{2}=0$, So $A-I$ has all eigenvalues $0$. Hence, all the
eigenvalues of $A$ are $1$ and $det(A)=1$. So, $V$ is of type $"+"$.

Furthermore, $n\equiv 0$ (mod $3$), then the images of the vectors
$w_{1}-w_{2},w_{1}-w_{3},...,w_{1}-w_{n-1},w_{n+1}$ form a basis for $V$. Let
us construct the Gram matrix: note that if $i\neq j$, then
$(w_{1}-w_{i},w_{1}-w_{j})=-1$ and $(w_{1}-w_{i},w_{1}-w_{j})=1$. Also,
$(w_{n+1},w_{n+1})=-1$ and $w_{n+1}$ is orthogonal to all other vectors. So,
the Gram matrix is of form

$A=\begin{pmatrix}-1&1&\cdots&1&0\\\ 1&-1&\cdots&1&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 1&1&\cdots&-1&0\\\
0&0&\cdots&0&-1\end{pmatrix}$

Note that, $rank(A-I)=2$, So $A-I$ has all eigenvalues, but $2$ eigenvalues
equal $0$. Furthermore, it has $2$ vectors corresponding to the eigenvalue
$1$. So, $Sp(A-I)=\\{0,0,...,0,1,1\\}$. Thus, $Sp(A)=\\{1,1,...,1,-1,-1\\}$
and $det(A)=1$. Therefore, $V$ is of type $"+"$.

Finally, if $(ij)\in S_{n}$, then $\psi(ij)=t(w_{i}-w_{j})$ and
$Q(w_{i}-w_{j})=Q(w_{i})+Q(w_{j})=1$. Thus, $\psi$ is the natural embedding of
$S_{n}$ into $O^{+,+}_{n-1}(3)$.

Let $n$ be a multiple of $3$. Suppose that $W$ is an $n$-dimensional
orthogonal space over the $3$-element field of type ”+”. Let $w_{1},...,w_{n}$
be a basis for $W$ with $Q(w_{i})=-1$, $(w_{i},w_{i})=1$, and
$(w_{i},w_{j})=0$, if $i\neq j$. Let $W^{\prime}=\\{x\in
W:(x,w_{1}+...+w_{n})=0\\}$. Then, $W^{\prime}$ is a degenerate space :
$w_{1}+...+w_{n}\in W^{{}^{\prime}}$. Thus, if we factor $W^{{}^{\prime}}$ by
the subspace generated by the vector $w_{1}+...+w_{n}$, we obtain the space
$V=W^{{}^{\prime}}/<w_{1}+...+w_{n}>$ which is an orthogonal space of
dimension $n-2$. Let $\tau\in S_{n}$. Then, let $\psi(\tau):W\rightarrow W$ be
a linear operator for which $\psi(\tau)(w_{i})=w_{\tau(i)}$. Furthermore, one
may restrict $\psi(\tau)$ on $V$. Note that if $(ij)\in S_{n}$, then
$\psi(ij)=t(v_{i}-v_{j})$ and $Q(v_{i}-v_{j})=1$. Moreover, calculations,
similar to the calculations carried out in the previous subcase show that the
space $V$ is of type ”-”. Thus, $\psi$ is the natural embedding of $S_{n}$
into $PO^{-,+}_{n-2}(3)$.

###  5.4 Orthogonal spaces over the 2-element field

Let $n$ be a multiple of $4$. Let $W$ be a symplectic space over the
$2$-element field with basis $w_{1},...,w_{n}$ such that $(w_{i},w_{j})=1$ if
$i\neq j$. Equip $W$ with an orthogonal form such that
$Q(w_{1})=...=Q(w_{n})=0$. Let $W^{\prime}=\\{x\in
W:(x,w_{1}+...+w_{n})=0\\}$. Then, $W^{\prime}$ is a degenerate space :
$w_{1}+...+w_{n}\in W^{{}^{\prime}}$. Moreover,
$Q(w_{1}+w_{2}+...+w_{n})=\frac{n(n-1)}{2}=0\in\mathbb{Z}_{2}$. Thus, if we
can factor $W^{{}^{\prime}}$ by the subspace generated by the vector
$w_{1}+...+w_{n}$ and obtain the space $V=W^{{}^{\prime}}/<w_{1}+...+w_{n}>$
which is an orthogonal space of dimension $n-2$. Let $\tau\in S_{n}$. Then,
let $\psi(\tau):W\rightarrow W$ be a linear operator for which
$\psi(\tau)(w_{i})=w_{\tau(i)}$. Furthermore, one may restrict $\psi(\tau)$ on
$V$. Note that if $(ij)\in S_{n}$, then $\psi(ij)=t(v_{i}+v_{j})$ and
$Q(v_{i}+v_{j})=1$. Thus, $\psi$ is the natural embedding of $S_{n}$ into
$O^{\epsilon}_{n-2}(2)$ Also, from the previous section, one may easily
deduce, that $\epsilon=-$, if $n\equiv 4$ (mod $8$) and the subspace is of
type $\epsilon=+$ if $8|n$.

Let $n$ be an integer congruent to $2$ modulo $4$. Let $W$ be a symplectic
space over the $2$-element field with basis $w_{1},...,w_{n}$ such that
$(w_{i},w_{j})=1$ if $i\neq j$. Equip $W$ with an orthogonal form such that
$Q(w_{1})=...=Q(w_{n})=\epsilon$. Note that
$W=<w_{1},w_{2}>\oplus<w_{1}+w_{2}+w_{3},w_{1}+w_{2}+w_{4}>\oplus...\oplus<w_{1}+w_{2}+...+w_{n-2}+w_{n-1},w_{1}+w_{2}+...+w_{n-2}+w_{n}>$.
Moreover, types of the spaces alternate. Thus, the spaces with $\epsilon=0$
and $\epsilon=1$ are not isomorphic. Furthermore, $S_{n}$ acts coordinate-wise
on both spaces, and a transposition $(ij)$ on $S_{n}$ has the same action as
the transvenction $t(v_{i}+v_{j})$. Therefore, we described a faithful action
of $S_{n}$ on both $O_{n}^{+}(2)$ and $O_{n}^{-}(2)$.

Let $n$ be an integer congruent to $1$ modulo $4$. Suppose that $W$ is an
orthogonal space of dimension $(n+1)$. Let $v_{1},...,v_{n+1}$ be a basis for
$W$ with $(v_{i},v_{j})=1$ if $v_{i}\neq v_{j}$ $Q(v_{1})=...=Q(v_{n+1})=0$.
The choice of $Q(v_{n+2})$ depends on the type of the orthogonal space. Note
that $Q(v_{1}+...v_{n+1})=\frac{(n+1)n}{4}=0\in Z_{2}$. Thus, if
$W^{{}^{\prime}}=\\{w_{1}+w_{2}+...+w_{n+1}\\}$ and
$V=W^{{}^{\prime}}/<w_{1}+...+w_{n+1}>$, then $V$ is an orthogonal space,
which inherits the dot product and quadratic form from $W$. Let $\tau\in
S_{n}$. Then, let $\psi(\tau):W\rightarrow W$ be a linear operator for which
$\psi(\tau)(w_{i})=w_{\tau(i)}$, for $i=\overline{1,...,n+1}$. Moreover, one
may restrict $\psi(\tau)$ on $V$. Note that $\psi(\tau)$ preserves the
orthogonal form, and for $i\neq j\in\\{1,...,n\\}$ $\psi(ij)=t(v_{i}+v_{j})$.
Thus, $\psi$ is the natural embedding of $S_{n}$ into $O_{n-1}^{\epsilon}(2)$.

##  6 Conclusions

To conclude the results obtained in the previous sections, we have computed
the following values for the groups in Fischer’s list.

######  Theorem 4.

The maximal symmetric subgroups of the groups from the Fischer’s group have
the following sizes:

  * 1.

$\varphi(S_{n})=n$;

  * 2.

$\varphi(Sp_{2}(2))=3$, $\varphi(Sp_{n}(2))=n+2$, if $n\geq 4$ is even;

  * 3.

$\varphi(U_{n}(2))=n$, if $n\leq 3$, $\varphi(U_{n}(2))=n+1$, if $n\geq 4$ is
odd, $\varphi(U_{n}(2))=n+2$, if $n\geq 4$ is even;

  * 4.

$\varphi({PO_{n}}^{+,+}(3))=n$, if $n=1$ (mod $3$), $n+1$, otherwise;

  * 5.

$\varphi({PO_{n}}^{-,+}(3))=n+2$, if $n=1$ (mod $3$), $n+1$ if $n=0$ (mod
$3$), $n$, if $n=2$ (mod $3$);

  * 6.

$\varphi({O_{4}}^{+}(2))=3$, $\varphi({O_{n}}^{+}(2))=n+1$, if $n=0$ (mod
$4$), $\varphi({O_{n}}^{+}(2))=n$, if $n=2$ (mod $8$) and
$\varphi({O_{n}}^{+}(2))=n+2$, if $n=6$ (mod $8$);

  * 7.

$\varphi({O_{2}}^{-}(2))=3$, $\varphi({O_{n}}^{+}(2))=n+1$, if $n=0$ (mod
$4$), $\varphi({O_{n}}^{+}(2))=n$, if $n=6$ (mod $8$) and
$\varphi({O_{n}}^{+}(2))=n+2$, if $n=2$ (mod $8$);

  * 8.

$\varphi(M(22))=10$, $\varphi(M(23))=12$, $\varphi(M(24))=12$.

In addition, in the previous section, for each classical group $G$ we have
built a faithful representation of $S_{\varphi(G)}$ into the corresponding
space of $G$, or into a smaller space, which is a direct geometrical
interpretation of the main results.

The theorem 3 is a direct corollary of this theorem. The groups listed in the
theorem 3 are candidates to admit standard Majorana representations. Note that
if for such a group $(G,D)$ there exists a monomorphism $\psi:G\rightarrow M$,
such that $\psi(D)\subseteq 2A$, then there is also a Majorana representation
of $G$ \- it suffices to take the subalgebra of the algebra $V_{M}$ generated
by the axes corresponding to the involutions from $\psi(D)$. Therefore, we
plan to find out the groups in this list, which admit a standard Majorana
representation, and among these groups find these which can be embedded in the
Monster.

##  7 Acknowledgments

I would like to thank my supervisor, A.A. Ivanov, for introducing me to the
area of research and for his useful advice.

## References

  * [1] M. Aschbacher, 3-Transposition groups, Cambridge Univ. Press, Cambridge, 1996.
  * [2] B. Fischer, Finite groups generated by 3-transpositions, University of Warwick, Lecture notes, 1969.
  * [3] B. Fischer, Finite groups generated by 3-transpositions, Invent. Math. 13 (1971) 232-246.
  * [4] C. Franchi, A.A. Ivanov, M. Mainardis, Standard Majorana representation of the symmetric groups, J. Alg. Combin. 44 (2016) 265-292.
  * [5] A.A. Ivanov, The Monster group and Majorana involutions, Cambridge Univ. Press, Cambridge, 2009.
  * [6] A.A. Ivanov, D.V. Pasechnik, A. Seress, S. Shpectorov, Majorana representations of the symmetric group of degree $4$, J. of Algebra 324(2010) 2432-2463.
  * [7] A.A. Ivanov, On Majorana representations of $A_{6}$ and $A_{7}$, Commun. in Math. Phys. 307(2011) 1-16.
  * [8] A.A. Ivanov, A. Seress, Majorana representations of $A_{5}$, Math. Z. 272(2012) 269-295.
  * [9] S.P. Norton, The Monster is fabulous, Cont. Math. 694(2017) 3-10.