# Commuting probability in algebraic groups

Shripad M. Garge  Department of Mathematics, Indian Institute of Technology,
Powai, Mumbai, 400 076 India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of commuting probability, $p(G)$, for an algebraic
group $G$. This notion is inspired by the corresponding notions in finite
groups and compact groups. The computation of $p(G)$ for reductive groups is
readily done using the notion of $z$-classes. We introduce two generalisations
of this relation, $iz$-equivalence and $dz$-equivalence. These notions lead us
naturally to the notion of a regular element in $G$. Finally, with the help of
this notion of regular elements, we compute $p(G)$ for a connected, linear
algebraic group $G$. We also compute the set of limit points of the numbers
$p(G)$ as $G$ varies over the classes of reductive groups, solvable groups and
nilpotent groups.

###### Key words and phrases:

Commuting probability, algebraic groups, $z$-classes, $G$-regular classes

##  1\. Introduction

_What is the probability that two randomly chosen elements of a finite group
commute?_

This was the question asked by Erdös and Turan in [2] where they formulated
the notion of commuting probability for a finite group. If $C$ denotes the set
of ordered pairs of commuting elements of a finite $G$ then $p(G)$, the
commuting probability of $G$, is defined to be the ratio $|C|/|G\times G|$. It
is proved in the same paper that this number is equal to $k/|G|$ where $k$ is
the number of conjugacy classes in $G$. Almost immediately, Gustafson ([5]),
generalised the notion of commuting probability to compact groups where he
used the Haar measure in the definition. There has been a flurry of activity
around this topic for many years since then.

The purpose of the present paper is to introduce the notion of commuting
probability in the case of algebraic groups. In algebraic geometry, the
natural measure of a certain algebraic set is its dimension. If $G$ is a
linear algebraic group then we define the set $C(G)$ to be the set of ordered
pairs of commuting elements in $G$, $C(G):=\\{(a,b)\in G\times G:ab=ba\\}$.
This is a Zariski closed subset of $G\times G$. We define the commuting
probability of $G$, denoted by $p(G)$, as follows

$p(G):=\frac{\dim(C(G))}{\dim(G\times G)}=\frac{\dim(C(G))}{2\dim(G)}.$

For a finite group $G$, which can be considered as an algebraic group with
$\dim(G)=0$, we define $p(G)=1$.

We note a basic lemma which computes the commuting probability of a direct
product.

######  Lemma 1.1.

If $G=H_{1}\times H_{2}$ then

$p(G)=\dfrac{\dim(H_{1})p(H_{1})+\dim(H_{2})p(H_{2})}{\dim(H_{1})+\dim(H_{2})}.$

###### Proof.

Since $G=H_{1}\times H_{2}$, it follows that $C(G)$ is isomorphic to
$C(H_{1})\times C(H_{2})$ and therefore
$\dim(C(G))=\dim(C(H_{1}))+\dim(C(H_{2}))$ which further equals
$2\dim(H_{1})p(H_{1})+2\dim(H_{2})p(H_{2})$. As
$\dim(G)=\dim(H_{1})+\dim(H_{2})$, the lemma now follows. ∎

If $G$ is a connected, linear algebraic group then so is $G\times G$ and then
the dimension of every proper closed subset of $G\times G$ is smaller than
$\dim(G\times G)$. Hence for a connected $G$, $p(G)=1$ if and only if $G$ is
abelian.

If $G$ is not connected, then $p(G)$ can be 1 without $G$ being abelian.
Consider $G=G_{1}\times G_{2}$ where $G_{1}$ is abelian and $G_{2}$ is a
finite non-abelian group. Then by the above lemma, $p(G)=1$ but $G$ is not
abelian.

To avoid such cases, we will work with connected groups in the remainder of
this paper. In any case, $p(G)=p(G^{0})$, so we are not losing out on the
generality. We also choose to work over $\mathbb{C}$, though the results
remain valid over an algebraically closed field, indeed over a perfect field.

We begin the study of $p(G)$ with the case of reductive groups in Section 2.
If $G$ is a reductive group of dimension $n$ and rank $r$ then we prove in
Theorem 2.1 that $p(G)=(n+r)/2n$. We also list the numbers $p(G)$ for all
simple groups. These calculations are used later while computing the limit
points of the set of the numbers $p(G)$.

The main tool used in the proof of Theorem 2.1 is the notion of $z$-classes
which is not useful in the case of a non-reductive connected $G$. Hence we
introduce the notions of $iz$-equivalence and $dz$-equivalence, generalising
the $z$-equivalence in the third section. We study the corresponding
equivalence classes and make some interesting observations, which may be of
independent interest. These notions lead us to the notion of a regular element
in $G$.

The dimension of the centralizer of a regular element in $G$ is called the
regular rank of $G$. We prove in Theorem 4.2 that if $G$ is a connected,
linear algebraic group with dimension $n$ and regular rank $r$ then
$p(G)=(n+r)/2n$. In this section, Section 4, we also list the numbers $p(U)$
where $U$ is the unipotent radical of a fixed Borel subgroup in a simple
algebraic group.

By the Theorem 4.2, $p(G)$ is a rational number $>1/2$. In Section 5, we
compute all rational numbers that can occur as $p(G)$ for various classes of
$G$, namely the reductive groups, solvable groups and nilpotent groups. As a
consequence, we compute the limit points of these numbers for all these above
classes. The set of limit points in all these cases is equal to the interval
$[1/2,1]$.

We then compare the results on $p(G)$ for algebraic groups with the
corresponding results in finite groups. We point out some similarities and
many differences between these two notions. This occupies the Section 6. In
the seventh section, we point out the compatibilities of $p(G)$ for a
semisimple group $G$ defined over $\mathbb{F}_{q}$ with the corresponding
finite groups of Lie type, in the asymptotic sense. We close the paper with
some concluding remarks in the last section. We also indicate some questions
that may of interest to the community.

## Acknowledgements

It is a pleasure to thank Dipendra Prasad, Saurav Bhaumik, Anuradha Garge and
Anupam Kumar Singh for many useful discussions. Anupam, in particular, has
been a catalyst for this paper in more than one ways.

##  2\. Reductive groups

Let $G$ be a complex reductive group. In this section, we compute the
commuting probability $p(G)$ of $G$ in terms of the dimension and rank of $G$.
The rank of a reductive group is the dimension of a maximal torus in $G$.
Since we compute dimensions, it will be sufficient to work with the group
$G(\mathbb{C})$. In the computation, we will use the notion of a $z$-class.
Let us first recall it.

Two elements $g,h\in G(\mathbb{C})$ are said to be _$z$ -equivalent_ if their
centralisers, $Z_{G(\mathbb{C})}(g)$ and $Z_{G(\mathbb{C})}(h)$, are conjugate
within $G(\mathbb{C})$. This is an equivalence relation and the corresponding
equivalence classes are called _$z$ -classes_. This equivalence is weaker than
the one given by the conjugacy relation. Each $z$-class is a union of certain
conjugacy classes.

It is known that if $G$ is a reductive group defined over $\mathbb{C}$ then
the number of $z$-classes in $G(\mathbb{C})$ is finite. The result is true in
a more general setting and we refer the reader to [3] for a detailed
discussion. For the present paper, the finiteness result over $\mathbb{C}$ is
sufficient.

######  Theorem 2.1.

Let $G$ be a complex reductive group with dimension $n$ and rank $r$. The
commuting probability of $G$, $p(G)$, is equal to

$\frac{n+r}{2n}=\frac{1}{2}+\frac{r}{2n}.$

###### Proof.

We will be working throughout the proof with the set of $\mathbb{C}$-rational
points without mentioning the field $\mathbb{C}$ explicitly. So, whenever we
talk about a subset of an algebraic set $X$, we will always mean a subset of
$X(\mathbb{C})$.

Let $\mathscr{C}$ denote the set of conjugacy classes in $G$ and let
$\mathfrak{c}$ be a conjugacy class in $G$. The set $C(G)$ is equal to the set

$\cup_{g\in G}\\{g\\}\times
Z_{G}(g)=\bigcup_{\mathfrak{c}\in\mathscr{C}}\bigg{(}\cup_{g\in\mathfrak{c}}\big{(}\\{g\\}\times
Z_{G}(g)\big{)}\bigg{)}=\bigcup_{\mathfrak{c}\in\mathscr{C}}C(G)_{\mathfrak{c}}.$

If $g,h\in\mathfrak{c}$ then $Z_{G}(g)$ and $Z_{G}(h)$ are conjugate in $G$,
in particular their dimensions are the same hence $\dim C(G)_{\mathfrak{c}}$
is equal to $\dim(G)$ for each $\mathfrak{c}\in\mathscr{C}$.

Let $\mathscr{Z}$ denote the set of $z$-classes in $G$. This is a finite set
and

$\mathscr{C}=\cup_{\mathfrak{z}\in\mathscr{Z}}\big{(}\mathfrak{z}/\mathfrak{c_{\mathfrak{z}}}\big{)}$

where the set $\mathfrak{z}/\mathfrak{c_{\mathfrak{z}}}$ is the set of
conjugacy classes in $\mathfrak{z}$.

Since $\mathscr{Z}$ is a finite set, there is one $z$-class in $G$ which is
dense in $G$. Indeed, the set of regular semisimple elements forms a $z$-class
in $G$ which is dense in $G$. We denote this $z$-class by $\mathfrak{z}_{0}$.
It then follows that
$\bigcup_{\mathfrak{c}\subseteq\mathfrak{z}_{0}}C(G)_{\mathfrak{c}}$ is dense
in $C(G)$ and hence $\dim(C(G))$ is equal to the dimension of
$\bigcup_{\mathfrak{c}\subseteq\mathfrak{z}_{0}}C(G)_{\mathfrak{c}}$.

Let us fix a conjugacy class $\mathfrak{c}$ in $\mathfrak{z}_{0}$. Then the
dimension of
$\bigcup_{\mathfrak{c}\subseteq\mathfrak{z}_{0}}C(G)_{\mathfrak{c}}$ is equal
to $\dim(\mathfrak{z}_{0}/\mathfrak{c})+\dim(G)$. Further,
$\dim(\mathfrak{z}_{0}/\mathfrak{c})$ is $\dim(G)-\dim(\mathfrak{c})$ which is
$\dim T$ for a maximal torus in $G$. Hence $\dim(C(G))=\dim(G)+\dim(T)=n+r$
and

$p(G)=\frac{n+r}{2n}=\frac{1}{2}+\frac{r}{2n}=\frac{1}{2}+\frac{\mathrm{rank}(G)}{2\dim(G)}.$

∎

######  Remark 2.2.

The analysis in the above proof implies that the dimension of the variety of
conjugacy classes in a reductive $G$ is equal to the dimension of a maximal
torus in $G$. This is a well-known fact, see [8, 6.4] for instance. However,
we have used $z$-classes because we use a generalisation of this notion in the
general case. The reason for that being that we have no information about the
dimension of the variety of conjugacy classes in $G$ in the general case. We
do not even know if the set of conjugacy classes forms a (quasi-projective)
variety.

######  Remark 2.3.

Using the above theorem, we note commuting probabilities of some groups,
including all simple algebraic groups.

  1. (1)

$p(GL_{n})=\dfrac{1}{2}+\dfrac{n}{2n^{2}}=\dfrac{1}{2}+\dfrac{1}{2n}=\dfrac{n+1}{2n}$,
for $n\geq 1$.

  2. (2)

$p(SL_{n})=\dfrac{1}{2}+\dfrac{n-1}{2(n^{2}-1)}=\dfrac{1}{2}+\dfrac{1}{2(n+1)}$,
for $n\geq 1$.

  3. (3)

$p(SO_{2n+1})=\dfrac{1}{2}+\dfrac{n}{2(2n^{2}+n)}=\dfrac{1}{2}+\dfrac{1}{2(2n+1)}$,
for $n\geq 2$.

  4. (4)

$p(Sp_{2n})=\dfrac{1}{2}+\dfrac{n}{2(2n^{2}+n)}=\dfrac{1}{2}+\dfrac{1}{2(2n+1)}$,
for $n\geq 3$.

  5. (5)

$p(SO_{2n})=\dfrac{1}{2}+\dfrac{n}{2(2n^{2}-n)}=\dfrac{1}{2}+\dfrac{1}{2(2n-1)}$,
for $n\geq 4$.

  6. (6)

$p(G_{2})=\dfrac{4}{7}$, $p(F_{4})=p(E_{6})=\dfrac{7}{13}$,
$p(E_{7})=\dfrac{10}{19}$ and $p(E_{8})=\dfrac{16}{31}$.

##  3\. Regular elements and regular rank

We notice that the following three properties of $z$-classes have been crucial
in the proof of Theorem 2.1.

  1. (1)

A $z$-class is a union of conjugacy classes,

  2. (2)

a reductive $G$ contains a dense $z$-class, and,

  3. (3)

the centralisers of two elements in a $z$-class (are conjugate within $G$,
hence are isomorphic as abstract groups, and hence they) have the same
dimensions.

######  Example 3.1.

A non-reductive $G$ need not contain a dense $z$-class. Let $U$ denote the
group of $3\times 3$ upper triangular matrices over $\mathbb{C}$. Then for

$g=\begin{pmatrix}1&a&b\\\ &1&c\\\ &&1\end{pmatrix},\hskip
14.22636ptZ_{U}(g)=\left\\{\begin{pmatrix}1&x&y\\\ &1&z\\\
&&1\end{pmatrix}:cx=az\right\\}.$

The group $U/Z(U)$ is abelian. If two centralisers in $U$ are conjugate then
their images in $U/Z(U)$ must be the same. Thus, the $z$-classes in $U$
correspond to the ratios $\frac{a}{c}\in(\mathbb{C}\cup{\infty})$ along with
the central $z$-class.

The dimension of each $z$-class is less than $3$. In particular, there is no
dense $z$-class in $U$.

Taking a cue from the above three properties we make the following two
definitions generalising the notion of $z$-equivalence.

######  Definition 3.2.

$(1)$ Two elements $x$ and $y$ in a group $G$ are called _$iz$ -equivalent_ if
the centralisers, $Z_{G}(x)$ and $Z_{G}(y)$, are isomorphic.

$(2)$ Two elements $x$ and $y$ in an algebraic group $G$ are called _$dz$
-equivalent_ if the centralisers, $Z_{G}(x)$ and $Z_{G}(y)$, have the same
dimension.

The $iz$-equivalence is also introduced by Dilpreet Kaur and Uday Bhaskar
Sharma. We are borrowing the name, $iz$-equivalence, with their kind
permission.

The above two relations are indeed equivalence relations. The corresponding
equivalence classes will be called $iz$-classes and $dz$-classes,
respectively. We have the following hierarchy in terms of the strengths of
these equivalences:

  * •

a $dz$-class is a union of $iz$-classes,

  * •

an $iz$-class is a union of $z$-classes and

  * •

a $z$-class is a union of conjugacy classes.

These equivalence relations are, in general, different as the following
examples show.

######  Example 3.3.

The $z$-class of central elements in a group is not always a single conjugacy
class.

######  Example 3.4.

We once again consider the group $U$ of $3\times 3$ upper triangular matrices
over $\mathbb{C}$. If we take

$x=\begin{pmatrix}1&1&1\\\ &1&0\\\ &&1\end{pmatrix},y=\begin{pmatrix}1&0&1\\\
&1&1\\\ &&1\end{pmatrix}\hskip 14.22636pt\mathrm{then}\hskip
14.22636ptZ_{U}(x)=\begin{pmatrix}1&a&b\\\ &1&0\\\
&&1\end{pmatrix},Z_{U}(y)=\begin{pmatrix}1&0&c\\\ &1&d\\\ &&1\end{pmatrix}$

as $a,b,c,d\in\mathbb{G}_{a}$. These centralisers are isomorphic to
$\mathbb{G}_{a}^{2}$. But they are not conjugate as their images in the
quotient $U/Z(U)$, which is abelian, are different one dimensional subgroups
of $U/Z(U)$.

Thus, $x,y$ are $iz$-equivalent but not $z$-equivalent.

######  Example 3.5.

Finally, we take $x$ to be a regular semisimple element in
$G=SL_{2}(\mathbb{C})$ and $y$ to be a regular unipotent element in the same
group. For instance, take

$x=\begin{pmatrix}\lambda&\\\ &\lambda^{-1}\end{pmatrix},\lambda\neq 1,\hskip
14.22636pt\mathrm{and}\hskip 14.22636pty=\begin{pmatrix}1&1\\\
&1\end{pmatrix}$

then the centraliser of $x$ is isomorphic to $\mathbb{G}_{m}$ and the
centraliser of $y$ is isomorphic to $\mathbb{G}_{a}$. Hence, $x$ and $y$ are
$dz$-equivalent, $\dim(\mathbb{G}_{m})=\dim(\mathbb{G}_{a})=1$, but not
$iz$-equivalent as $\mathbb{G}_{m}$ and $\mathbb{G}_{a}$ are not isomorphic.

######  Remark 3.6.

Note that in the Example 3.4, the two centralisers are conjugate in $SL_{3}$,
a bigger group than $U$.

There are two ways to see this. The elements $x$ and $y$ are themselves
conjugate in $SL_{3}$ and hence their centralisers are conjugate in $SL_{3}$
which, in this case, agree with their centralisers in $U$. Alternatively, the
two centralisers correspond to two different simple systems of roots in the
root system $\Phi$ of $SL_{3}$, hence they are conjugate by a Weyl group
element.

If we were to consider only abstract groups, then the celebrated HNN-theory
tells us that two isomorphic centralisers in a group become conjugate in a
bigger group.

Emboldened by the above remark and especially the Example 3.4, we ask

######  Question 3.7.

If $G$ is a linear algebraic group and $H_{1}$, $H_{2}$ are two isomorphic
subgroups of $G$, then is there a linear algebraic group containing $G$ in
which $H_{1}$ and $H_{2}$ become conjugate?

Equivalently, if $H_{1},H_{2}\subseteq G$ are isomorphic then do we have an
embedding of $G$ in some $GL_{n}$ such that $H_{1},H_{2}$ become conjugate in
$GL_{n}$?

Such a question for $iz$-equivalence vis-á-vis $z$-equivalence would be
difficult to consider because, unlike Example 3.4, the centralisers may change
in a bigger group. We therefore only ask how the $iz$-equivalence in $G$
behaves with respect to the embeddings of $G$ in bigger groups. Do
$iz$-equivalent elements in $G$ remain $iz$-equivalent in all such embeddings?
Do they become $z$-equivalent in some embedding? What would be the situation
for finite groups of Lie type?

Such questions will not make sense for other equivalences. We may have a
semisimple element $z$-equivalent to a unipotent element, in an abelian group
for instance, but these two elements will never be conjugate in a bigger
group. Indeed, they may not remain $z$-equivalent in a bigger group.

Consider, for instance, the two distinct elements $1=x,y$ in
$B_{2}(\mathbb{F}_{2})$, the Borel subgroup in $GL_{2}(\mathbb{F}_{2})$. Since
the group $B_{2}(\mathbb{F}_{2})$ is abelian, $x$ and $y$ are $z$-equivalent
but they don’t remain $z$-equivalent in bigger groups, in
$B_{2}(\mathbb{F}_{4})$ for instance.

Similarly, $x$ and $y$ in Example 3.5 can never be $iz$-equivalent in a bigger
group.

######  Remark 3.8.

The number of $iz$-classes in group $U$ above is finite. However, we do not
know if this holds in general. If we knew that the number of $iz$-classes in
an algebraic group is finite, at least, for a class of groups (other than the
reductive groups), then we would be able to generalise the proof of Theorem
2.1 for this class of groups. However, we have no such information at present.
We do not even know if there is a dense $iz$-class in a class of algebraic
groups (other than the reductive groups).

Hence we consider the $dz$-classes, whose number is finite for every algebraic
group. It also follows that every algebraic group contains a dense $dz$-class.

######  Definition 3.9.

Let $G$ be a linear algebraic group. An element $g\in G$ is called regular if
the dimension of its centraliser, $Z_{G}(g)$, is minimum among such
dimensions.

Equivalently, an element $g\in G$ is regular if its conjugacy class has the
largest possible dimension.

We note that the set of regular elements in a $dz$-class, which is open (and
hence dense) in $G$ and the centralisers of two elements in this class have
the same dimensions. These were the three properties that we had listed at the
beginning of this section.

######  Definition 3.10.

Let $G$ be a linear algebraic group. The _regular rank_ of $G$ is the
dimension of the centraliser of a regular element in $G$.

We are now ready to prove the general version of Theorem 2.1.

##  4\. Non-reductive groups

In this section, $G$ is a connected, linear algebraic group defined over
$\mathbb{C}$. The set of regular elements in $G$ is denoted by $G_{reg}$.

We note some basic results about the regular rank. The proofs are omitted.

######  Remark 4.1.

  1. (1)

If $G$ is reductive then the regular rank of $G$ is the same as its rank, the
dimension of a maximal torus in $G$.

  2. (2)

If $G_{1}\subseteq G_{2}$ then the regular rank of $G_{1}$ is less than or
equal to that of $G_{2}$.

  3. (3)

The regular rank of $G_{1}\times G_{2}$ is the sum of the regular ranks of
$G_{i}$.

  4. (4)

The regular rank of a semidirect product $G_{1}\ltimes G_{2}$ need not be
equal to the sum of the regular ranks of $G_{i}$. We will see an example of
this, a Borel in a semisimple group, in Lemma 4.4.

######  Theorem 4.2.

Let $G$ be a complex, connected, linear algebraic group. If the dimension of
$G$ is $n$ and the regular rank of $G$ is $r$ then the commuting probability
of $G$ is

$\frac{n+r}{2n}=\frac{1}{2}+\frac{r}{2n}.$

###### Proof.

The proof develops on the similar lines as the proof of Theorem 2.1.

We note that $C(G)$ is equal to the union
$\cup_{\mathfrak{c}\in\mathscr{C}}C(G)_{\mathfrak{c}}$ where $\mathscr{C}$ is
the set of conjugacy classes in $G$ and $\dim(C(G)_{\mathfrak{c}})=\dim(G)$
for every $\mathfrak{c}\in\mathscr{C}$.

Since the set $G_{reg}$ is dense in $G$ and is a union of conjugacy classes in
$G$, we have that $\dim(C(G))$ is equal to
$\dim(G_{reg}/\mathfrak{c})+\dim(G)$ where the set $G_{reg}/\mathfrak{c}$ is
the set of conjugacy classes in $G_{reg}$. The dimension of
$G_{reg}/\mathfrak{c}$ is equal to $\dim(G)-\dim(\mathfrak{c})=\dim Z_{G}(g)$,
where $g$ is a fixed element in $\mathfrak{c}$. Thus,
$\dim(G_{reg}/\mathfrak{c})$ is the regular rank of $G$. Hence
$\dim(C(G))=n+r$ and

$p(G)=\frac{n+r}{2n}=\frac{1}{2}+\frac{r}{2n}.$

∎

######  Remark 4.3.

For a connected linear algebraic group $G$, if the set $\mathscr{C}$ of
conjugacy classes forms a variety in a natural way then it follows from the
above proof that $\dim(\mathscr{C})$ is the same as the regular rank of $G$.

We now compute the regular ranks of certain subgroups of semisimple groups.

######  Lemma 4.4.

Let $G$ be a complex semisimple group. We fix a Borel subgroup $B$ in $G$ and
let $U$ be the unipotent radical of $B$. Then the regular rank of $U$ is equal
to the rank of the group $G$.

The regular rank of a parabolic in $G$ is equal to the rank of $G$. In
particular, the regular rank of $B$ is equal to the rank of $G$.

###### Proof.

Let $u$ be a regular unipotent element of $G$ which belongs to $U$. It is
proved by Steinberg, [8, $\S 4$], that such elements exist. The dimension of
$Z_{G}(u)$ is equal to the rank of $G$ which is bigger than or equal to the
regular rank of $U$. Further, since $u$ is in a unique Borel subgroup of $G$,
it follows that $Z_{G}(u)=Z_{U}(u)$. Hence the regular rank of $U$ is equal to
the rank of $G$.

If $P$ is a parabolic in $G$ then, up to conjugacy in $G$, $U\subset P\subset
G$. It follows, from Remark 4.1 (2), that the regular rank of $P$ is also
equal to the rank of $G$. ∎

######  Corollary 4.5.

Let $G$ be as in the above lemma with dimension $n$ and rank $r$, and let $U$
be as in the above lemma. Then

$p(U)=\frac{(\frac{n-r}{2})+r}{n-r}=\frac{1}{2}+\frac{r}{n-r}.$

###### Proof.

Follows from the above two results and that $2\dim(U)+r=n$. ∎

######  Remark 4.6.

Using the above result, we note commuting probabilities of the unipotent
radicals of Borel subgroups of all simple groups. For a simple $G$, the
corresponding unipotent radical is denoted by $U(G)$.

  1. (1)

$p(U(SL_{n}))=\dfrac{1}{2}+\dfrac{n-1}{n^{2}-n}=\dfrac{1}{2}+\dfrac{1}{n+1}$,
for $n\geq 1$.

  2. (2)

$p(U(SO_{2n+1}))=\dfrac{1}{2}+\dfrac{n}{2n^{2}}=\dfrac{1}{2}+\dfrac{1}{2n}$,
for $n\geq 2$.

  3. (3)

$p(U(Sp_{2n}))=\dfrac{1}{2}+\dfrac{n}{2n^{2}}=\dfrac{1}{2}+\dfrac{1}{2n}$, for
$n\geq 3$.

  4. (4)

$p(U(SO_{2n}))=\dfrac{1}{2}+\dfrac{n}{2n^{2}-2n}=\dfrac{1}{2}+\dfrac{1}{2n-2}$,
for $n\geq 4$.

  5. (5)

$p(U(G_{2}))=\dfrac{2}{3}$, $p(U(F_{4}))=p(U(E_{6}))=\dfrac{7}{12}$,
$p(U(E_{7}))=\dfrac{5}{9}$ and $p(U(E_{8}))=\dfrac{27}{50}$.

##  5\. Limit points

It follows from Theorem 4.2 that $\frac{1}{2}<p(G)\leq 1$. We also have that
$p(G)=1$ if and only if $G$ is abelian. In this section, we investigate the
possible values of $p(G)$ in $[\frac{1}{2},1]$ as $G$ varies over all linear
algebraic groups. We will also compute the limit points of these numbers.

######  Lemma 5.1.

  1. (1)

An $\alpha\in[\frac{1}{2},1]$ is $p(G)$ for a simple $G$ if and only if
$\alpha=\frac{1}{2}+\frac{1}{2m}$ for some $m>1$.

  2. (2)

The number of simple groups $G$ with $p(G)>p/q>1/2$ is finite.

  3. (3)

The set of these numbers, from $(1)$ above, has only one limit point which is
$\frac{1}{2}$.

###### Proof.

The first part of the lemma is evident from Remark 2.3.

The inequality $p(G)>p/q$ gives $1/2+r/2n>p/q$ which gives $r/n>(2p-q)/q>0$
and $n/r<q/(2p-q)$. The number $n/r$ is an integer for a simple algebraic
group and it follows, from Remark 2.3 for instance, that there are only
finitely many simple $G$ whose $n/r$ is bounded above, for every bound. Thus,
there are only finitely many simple $G$ with $p(G)>p/q>1/2$.

Finally, from (2) it follows that $1/2$ is the only possible limit point of
the set of numbers $p(G)$ where $G$ varies over simple algebraic groups. We
see that $1/2$ is indeed a limit point, $\lim_{n\to\infty}p(SL_{n})=1/2$. ∎

If we consider the reductive groups, instead of only the simple ones, then the
above lemma does not hold. For example, if $G=GL_{2}\times\mathbb{G}_{m}^{a}$
then $p(G)=\frac{3+a}{4+a}$ and the limit of these numbers is $1$! In fact, we
have the following result:

######  Proposition 5.2.

Every rational number from the set $(\frac{1}{2},1]$ is $p(G)$ for some
reductive group $G$.

###### Proof.

Assume that we have a rational number $1/2<p/q\leq 1$. Choose an $n$ with
$1/2<(n+1)/2n\leq p/q$. This is possible as $\\{(n+1)/2n\\}$ is a decreasing
sequence whose limit is $1/2$.

Let $G=GL_{n}^{a}\times\mathbb{G}_{m}^{b}$ where $a=q-p$ and
$b=(2n^{2}p-n^{2}q-nq)/2$.

Here, $a\geq 0$ as $p/q\leq 1$. Further, $b$ is an integer as $n^{2}-n$ is
always even and $b\geq 0$ if and only if $2np-(n+1)q\geq 0$ which holds as
$(n+1)/2n\leq p/q$.

The rank of $G$ is equal to $an+b$ and the dimension is equal to $an^{2}+b$.
Then

$p(G)=\frac{(an^{2}+b)+(an+b)}{2(an^{2}+b)}=\frac{(n^{2}-n)p}{(n^{2}-n)q}=\frac{p}{q}.$

∎

We have the same result for nilpotent groups.

######  Proposition 5.3.

Every rational number from the set $(\frac{1}{2},1]$ is $p(G)$ for some
nilpotent group $G$.

###### Proof.

The proof follows exactly in the similar way as the proof of the above
proposition. The role of $\mathbb{G}_{m}$ in the above proof will be played by
$\mathbb{G}_{a}$ here. ∎

######  Corollary 5.4.

  1. (1)

$\\{p(G):G$ is reductive$\\}=\\{p(G):G$ is nilpotent$\\}=\\{p(G):G$ is
solvable$\\}=\\{p(G):G$ is a linear algebraic
group$\\}=\mathbb{Q}\cap(1/2,1]$.

  2. (2)

Every $\alpha\in[1/2,1]$ is a limit point of each of the sets in $(1)$ above.

##  6\. Comparisons with the finite case

As mentioned in the introduction, the notion of $p(G)$ for finite groups has
been studied extensively. A review of whether the analogous results hold for
our notion of $p(G)$ is in order.

###  6.1. Simple groups

If $G$ is a non-abelian finite simple group then $p(G)\leq 1/12$, as was
observed by J. Dixon ([4, Introduction]). The probabilities $p(G)$ for simple
algebraic groups are also well-behaved, Lemma 5.1.

###  6.2. Direct products

If $G_{1}$ and $G_{2}$ are finite groups then $p(G_{1}\times
G_{2})=p(G_{1})p(G_{2})$. However, in the case of algebraic groups
$p(G_{1}\times G_{2})$ is almost never equal to $p(G_{1})p(G_{2})$. One way to
understand this is that $p(G)$ for an algebraic group is always bounded below
by $1/2$. The multiplicativity of $p(G)$ for direct products of algebraic
groups would say that for a non-abelian $G$ and a sufficiently large integer
$r$, $p(G^{r})=p(G)^{r}<1/2$ giving a contradiction.

###  6.3. Solvable groups and non-solvable groups

If $G$ is a finite group with $p(G)>3/40$ then Guralnick and Robinson prove
that $G$ is either solvable or is isomorphic to $A_{5}\times T$ for an abelian
group $T$ ([4, Theorem 11]). Thus, the numbers $p(G)$ for non-solvable groups
are bounded above by $3/40$, except for $p(A_{5})=1/12$. In contrast, the
numbers $p(G)$ for non-abelian reductive (hence non-solvable) algebraic groups
$G$ take all rational values in $(1/2,1)$, Corollary 5.4 (1).

###  6.4. Limit points

If $G$ is a finite non-abelian group then $p(G)\leq 5/8$ ([5, Introduction]).
Further, there are gaps within the numbers $p(G)$. However, the numbers $p(G)$
for algebraic groups take all rational values in $(1/2,1]$, Corollary 5.4 (1).
As a result, every real number in $[1/2,1]$ is a limit point of the set
$\\{p(G):G$ is algebraic$\\}$ whereas the set of limit points in the finite
case is a nowhere dense set of rational numbers ([1, Corollary 1.3]).

###  6.5. Isoclinism

Two groups $G$ and $H$ are called isoclic if there are isomorphisms
$\phi:G/Z(G)\to H/Z(H)$ and $\psi:G^{\prime}\to H^{\prime}$ such that the
diagram

${G/Z(G)\times G/Z(G)}$${H/Z(H)\times
H/Z(H)}$${G^{\prime}}$${H^{\prime}}$$\scriptstyle{\phi\times\phi}$$\scriptstyle{\alpha_{G}}$$\scriptstyle{\alpha_{h}}$$\scriptstyle{\psi}$

is commutative, where $\alpha_{G}(aZ(G),bZ(G))=aba^{-1}b^{-1}$ and
$\alpha_{H}$ is defined analogously.

It is proved in [6, Lemma 2.4] that if $G$ and $H$ are finite isoclinic groups
then $p(G)=p(H)$.

We can define isoclinism in exactly the same way as above for algebraic
groups. Then $GL_{n}$ and $SL_{n}$ are isoclinic groups with different
commuting probabilities.

##  7\. Compatibility with the finite groups of Lie type

It seems from the above section that there are more differences than
similarities when we compare $p(G)$ for algebraic groups with the $p(G)$ for
finite groups. This is not surprising as our definition of $p(G)$ for
algebraic groups uses the dimension, which is additive for direct products,
and then we take the ratios of the dimensions. This is the central reason for
many differences noted in the previous section. In spite of this, we contend
that our notion of $p(G)$ is a natural notion by comparing it with $p(G)$ of
the finite groups of Lie type.

Clearly, the characteristic of the field $\mathbb{C}$ played no role in the
proofs of our results. Hence the results stand good for groups defined over an
algebraically closed field. In fact, if $G$ is a connected, linear algebraic
group defined over a perfect field $k$ then the dimension of $G$ and the
regular rank of $G$ is defined over $k$. Hence the results proved in the paper
hold good over the field $k$ as well.

Let $k=\mathbb{F}_{q}$ be the finite field with $q$ elements and let $G$ be a
semisimple group defined over $k$ of dimension $n$ and rank $r$. We recall a
result of Steinberg.

######  Theorem 7.1 (Steinberg).

[7, 14.11] The number of semisimple conjugacy classes in $G(k)$ is equal to
$q^{r}$.

Further, the number of all conjugacy classes in $G(\mathbb{F}_{q})$ is
$O(q^{r})$ and the cardinality of $G(\mathbb{F}_{q})$ is $O(q^{n})$. The
commuting probability of the finite group $G(\mathbb{F}_{q})$ is therefore
$O(q^{r})/O(q^{n})=O(q^{n+r})/O(q^{2n})$ which is compatible with the
commuting probability of the algebraic group $G$.

##  8\. Concluding remarks

###  8.1. Non-affine groups

The set $C(G)$ makes sense even when $G$ is not an affine algebraic group and
hence we can define $p(G)$ in exactly the same way for a non-affine $G$ as in
the affine case. Further, the notion of a regular element also makes sense for
an algebraic group $G$, the set of regular elements is a dense subset of $G$
and hence our main result, Theorem 4.2, remains valid that $p(G)=(n+r)/2n$.
However, at this point, we do not have any computation of $p(G)$ for a non-
affine algebraic group, except when $G$ is an abelian variety with $p(G)=1$.

###  8.2. Further possibilities

While the set $C(G)$ of commuting pairs in a reductive group $G$ has been
studied previously, no one seems to have introduced the notion of the
commuting probability of $G$. The notion in the case of finite groups has many
applications and has been extensively investigated. We hope that this notion
also gets investigated.

There are many ways in which the probabilistic studies of algebraic groups can
be done. We list some possibilities below. It is our hope that they will also
be taken up.

######  Question 8.1.

We prove in Section 5 that every rational number in $(1/2,1]$ is $p(G)$ for a
linear algebraic group $G$. However, in the proof we have used groups that
have a large abelian direct factor. What are the value sets of $p(G)$ where
$G$ has no abelian direct factor? What are the limit points of this set?

######  Question 8.2.

If $H$ is an algebraic subgroup of $G$ then we can define $p(G,H)$ as

$p(G,H)=\frac{\dim(C(G,H))}{\dim(G\times H)}$

where $C(G,H)=\\{(g,h)\in G\times H:gh=hg\\}$. This $p(G,H)$ gives the
probability that two randomly chosen elements $g\in G$ and $h\in H$ commute
with each other.

Following the sections 2–4, it does not seem difficult to compute $p(G,H)$
when $H$ is a parabolic in a reductive $G$. It would be interesting to study
this notion in general, especially when $G$ and $H$ are unipotent. We have no
information on the difficulty level of this question in the unipotent case.

## References

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