# Roots of identity in finite groups of Lie type

Saikat Panja<EMAIL_ADDRESS>Harish-Chandra Research Institute- Main
Building, Chhatnag Rd, Jhusi, Uttar Pradesh 211019, India

###### Abstract.

Given an integer $M\geq 2$, we deploy the generating function techniques to
compute the number of $M$-th roots of identity in some of the well-known
finite groups of Lie type, more precisely for finite general linear groups,
symplectic groups, orthogonal groups of all types and unitary groups over
finite fields of odd characteristics.

###### Key words and phrases:

word maps, finite groups of Lie type, $M$-th root

###### 2020 Mathematics Subject Classification:

20G40, 20D06

The author is supported by a PDF-Math fellowship from the Harish-Chandra
Research Institute

##  1\. Introduction

A significant amount of modern mathematics is centred around finding solutions
to equations in objects belonging to different categories, for example, the
category of groups, the category of associative algebras, the category of Lie
algebras _etc._. In the world of groups, these are known as _word problems_.
It has come to the consideration of many great mathematicians of the current
century, especially after the settlement of the long-standing Ore’s conjecture
(which states that every element of a finite non-abelian simple group is a
commutator). Let us recall the word problem in the context of group theory in
brief. Fix a free group $\mathscr{F}_{n}$ on $n$ generators. Then for any word
$\omega\in\mathscr{F}_{n}$ and a group $G$, we get a map
$\widetilde{\omega}:G^{n}\longrightarrow G$, by means of evaluations. Several
astonishing results have been established regarding these maps. A typical
_word problem_ asks if the map induced by the given word is surjective on the
group or not. The other related problem i,e, _Waring-type problem_ asks for
the existence of $N(\omega)\in\mathbb{N}$ such that
$\langle\widetilde{\omega}(G)\rangle=\widetilde{\omega}(G)^{N(\omega)}$. The
most surprising result in this direction states that, for a given word
$\omega$, there exists $N_{\omega}\in\mathbb{N}$ such that for all finite non-
abelian simple groups of size greater or equal to $N_{w}$, we have
$N(\omega)=3$, see the great work [31] by Shalev. Later he together with
Larsen and Tiep improved the result by proving $N(\omega)=2$, in [18] and
[20]. Unfortunately the number $N(\omega)$ can not be reduced further. An easy
example arises out of the power map, which is induced by the word $\
omega=x^{M}\in\mathscr{F}_{1}$, for some integer $M\geq 2$. The Waring-type
problems have also been studied in case of Lie groups and Chevalley groups in
[13], unipotent algebraic groups in [17], residually finite groups in [19],
$p$-adic and Adelic groups in [1] _et cetera_. For a survey of these results
and further problems in the context of group theory, we refer the reader to
the excellent survey due to Shalev [31] and the references therein.

There have been studies to find the number of elements which are images of the
map $x^{M}$ in classical groups. This was done for the finite general linear
groups in [16], for the finite symplectic and orthogonal groups in [29] and
for finite unitary groups in [30]. The exceptional cases remain open. After
knowing that an equation has a solution in the concerned object, it is
desirable to ask how many solutions exist in this case. Some authors also call
them as the problem of finding fibres of a word map. The enumeration of the
solutions of $x^{p}=1$ has been carried out for symmetric and alternating
groups in many works like [4], [28], [15], [14] _et cetera_. The problem of
finding fibres of $x^{M}=1$, has been studied by several authors, for example,
proportions of elements in finite groups of Lie type has been estimated in
[27], algorithms to identify abundant $p$-singular elements in finite
classical groups have been studied in [26], Proportions of $p$-regular
elements in finite classical groups have been studied in [2], abundant
$p$-singular elements in finite classical groups have been studied in [25] _et
ceetra_. In this paper we derive the generating functions for the probability
of an element satisfying $x^{M}=1$, where $M$ is an integer greater than $2$.
We call these elements to be _roots of identity_. Note that if an element is a
root of identity, then so are all its conjugates. Thus we first find the
conjugacy class representatives which are $M$-th root of identity. In this
regard, we brush through a brief description of the conjugacy classes in
Section 2. Hereafter in Section 3, we first compute the roots of identity
which are either semisimple or unipotent. Finally using the Jordan
decomposition of an element, in Section 4, we derive the generating functions
for the number of $M$-th roots of identity in finite classical groups of Lie
type; more precisely in the general linear group, the symplectic group, the
orthogonal group and the unitary group. Finally in Section 5, we derive an
exact formula for the probability of a matrix to be $M$-th root of identity in
the case of the general linear group and the symplectic group, where $M$ is an
odd non-defining prime and the defining prime satisfies $q\equiv-1\pmod{M}$.

### Running assumption

Throughout the article it will be assumed that the finite fields under
consideration are of odd characteristics.

### Notation

The set of units in $\mathbb{Z}/n\mathbb{Z}$ will be denoted by
$\mathbb{Z}/n\mathbb{Z}^{\times}$. By the notation $e(n)$, we will always
denote the order of $q$ in the multiplicative group
$\mathbb{Z}/n\mathbb{Z}^{\times}$. The Euler’s totient function will be
denoted by $\phi(n)$. We denote the set of irreducible monic polynomials over
$\mathbb{F}_{q}$ by $\Phi$. Let us also fix the notation for the set of all
partitions of $n$ to be $\Lambda^{n}$. By the symbol
$\left(\dfrac{u}{q}\right)_{i}$, we denote the quantity
$\left(1-\dfrac{u}{q}\right)\left(1-\dfrac{u}{q^{2}}\right)\ldots\left(1-\dfrac{u}{q^{i}}\right)$.
For a partition $\lambda\vdash n$, the number of occurrences of $i$ will be
denoted as $m_{i}(\lambda)$. The transpose of a partition $\lambda$ will be
denoted as $\lambda^{\prime}$. The number of partitions of a positive integer
$n$ will be denoted as $\mathbf{P}(n)$ and those with lesser than or equal to
equal to $m$ many (not necessarily distinct) parts will be denoted as
$\mathbf{P}_{m}(n)$.

##  2\. The groups under consideration

In this section, we briefly recall the groups. Note that if an element is an
$M$-th root of identity, then so are its conjugates. Hence we embark to find
first the conjugacy classes which are $M$-th root of identity. This makes it
necessary for us to know about the conjugacy classes and the centralizer of an
element. We will follow the treatment of [11], [23], [36] and, [5]. For the
case of the general linear group, it is easy to find the conjugacy
representative for the conjugacy classes, but for other concerned groups, they
are a little tricky. One may look into [34], [33] and, [35], to handle the
other cases. Without further delay let us start with the subsection where we
talk about the general linear group.

###  2.1. General linear group

Given a $n$-dimensional vector space $V$ over $\mathbb{F}_{q}$, the set of all
automorphisms of $V$ is denoted by $\mathrm{GL}(V)$. After fixing an
appropriate basis, $\mathrm{GL}(V)$ can be identified with
$\mathrm{GL}_{n}(q)$. The conjugacy classes of $\mathrm{GL}_{n}(q)$ is
determined by functions $\lambda:\Phi\longrightarrow\Lambda$ by
$f\mapsto\lambda_{f}$ such that $\sum\limits_{f}\deg(f)\cdot|\lambda_{f}|=n$.
The combinatorial data attached via this correspondence, for an element
$\alpha\in\mathrm{GL}_{n}(q)$ will be denoted by $\Delta(\alpha)$. The
centralizer size is given by

$\displaystyle\prod\limits_{\varphi\in\Delta(\alpha)}\left(q^{\deg\varphi\cdot\sum\limits_{i}(\lambda_{i,\varphi}^{\prime})^{2}}\prod_{i\geq
1}\left(\dfrac{1}{q^{\deg\varphi}}\right)_{m_{i}(\lambda_{\varphi})}\right).$

This quantity sometimes can be shortened using a general expression, which we
recall from [6, Theorem 2] below;

$\displaystyle
c_{\mathrm{GL},\varphi,q}(\lambda)=\left(q^{\deg\varphi\cdot\sum\limits_{i}(\lambda_{i}^{\prime})^{2}}\prod_{i\geq
1}\left(\dfrac{1}{q^{\deg\varphi}}\right)_{m_{i}(\lambda_{\varphi})}\right).$

###  2.2. Symplectic and orthogonal groups

####  2.2.1. Symplectic group

Let $V$ be a vector space of dimension $2n$ over $\mathbb{F}_{q}$. There is a
unique non-degenerate alternating bilinear form on $V$. We consider the form
given by

$\left<(x_{i})_{i=1}^{2n},(y_{j})_{j=1}^{2n}\right>=\sum_{j=1}^{n}x_{j}y_{2n+1-j}-\sum_{i=0}^{n-1}x_{2n-i}y_{i+1}.$

The symplectic group is the subgroup of $\mathrm{GL}(V)$ consisting of those
elements which preserve this alternating form on $V$. By fixing an appropriate
basis, the matrix of the form is $J=\begin{pmatrix}0&\Pi_{n}\\\
-\Pi_{n}&0\end{pmatrix}$ where $\Pi_{n}=\begin{pmatrix}0&0&\cdots&0&1\\\
0&0&\cdots&1&0\\\ \vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
1&0&\cdots&0&0\end{pmatrix}$ and

$\mathrm{Sp}_{2n}(q)=\\{A\in\mathrm{GL}_{2n}(q)\mid{}^{t}\\!AJA=J\\}.$

Since all alternating forms are equivalent over $\mathbb{F}_{q}$, the
symplectic groups obtained with respect to different forms are conjugate
within $\mathrm{GL}(2n,q)$.

####  2.2.2. Orthgonal Groups

Let $V$ be an $m$-dimensional vector space over a finite field
$\mathbb{F}_{q}$. Then there are at most two non-equivalent non-degenerate
quadratic forms on $V$. The orthogonal group consists of elements of
$\mathrm{GL}(V)$ which preserve a non-degenerate quadratic form $Q$.

When $m=2n$ for some $n\geq 1$, up to equivalence there are two such forms
denoted as $Q^{+}$ and $Q^{-}$. These are as follows. Fix $a\in\mathbb{F}_{q}$
such that $t^{2}+t+a\in\mathbb{F}_{q}[t]$ is irreducible. Then the two non-
equivalent forms are given by

  1. (1)

$Q^{+}(x_{1},\ldots,x_{m})=x_{1}x_{2}+x_{3}x_{4}+\cdots+x_{2n-1}x_{2n}$, and

  2. (2)

$Q^{-}(x_{1},\ldots,x_{m})=x_{1}^{2}+x_{1}x_{2}+ax_{2}^{2}+x_{3}x_{4}+\cdots+x_{2n-1}x_{2n}$.

The orthogonal group preserving $Q^{+}$ will be denoted as
$\mathrm{O}^{+}_{m}(q)$, and the orthogonal group preserving $Q^{-}$ will be
denoted as $\mathrm{O}^{-}_{m}(q)$.

When $m=2n+1$, for $q$ even there is only one (up to equivalence) quadratic
form, namely
$Q(x_{1},\ldots,x_{m})=x_{1}^{2}+\sum\limits_{i=1}^{n}x_{2i}x_{2i+1}$ and
hence there is only one (up to conjugacy) orthogonal group. If $q$ is odd,
then up to equivalence, there are two non-degenerate quadratic forms. But,
these two forms give isomorphic orthogonal groups. We take
$Q(x_{1},\ldots,x_{m})=x_{1}^{2}+\cdots+x_{m}^{2}$. Thus, in case $m=2n+1$, up
to conjugacy, we have only one orthogonal group. This will be denoted as
$\mathrm{O}(m,q)$.

As it is common in literature, we will use the notation
$\mathrm{O}^{\epsilon}_{m}(q)$ to denote any of the orthogonal group above
where $\epsilon\in\\{~{},+,-\\}$. With respect to an appropriate basis, we
will fix the matrices of the symmetric bilinear forms (associated to the
quadratic forms $Q^{\epsilon}$) as follows:

$J_{0}=\begin{pmatrix}0&0&\Pi_{n}\\\ 0&\alpha&0\\\
\Pi_{n}&0&0\end{pmatrix},J_{+}=\begin{pmatrix}0&\Pi_{n}\\\
\Pi_{n}&0\end{pmatrix},J_{-}=\begin{pmatrix}0&0&0&\Pi_{n-1}\\\ 0&1&0&0\\\
0&0&-\delta&0\\\ \Pi_{n-1}&0&0&0\end{pmatrix}$

where
$\alpha\in\mathbb{F}_{q}^{\times},\delta\in\mathbb{F}_{q}\setminus\mathbb{F}_{q}^{2}$,
and $\Pi_{l}=\begin{pmatrix}0&0&\cdots&0&1\\\ 0&0&\cdots&1&0\\\
\vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
1&0&\cdots&0&0\end{pmatrix}$, an $l\times l$ matrix. Then, the orthogonal
group in matrix form is

$\mathrm{O}^{\epsilon}_{m}(q)=\\{A\in\mathrm{GL}_{m}(q)\mid{}^{t}\\!{A}J_{\epsilon}A=J_{\epsilon}\\}.$

Adapting the notations of [8], we define the type of an orthogonal space as
follows.

_The type of an orthogonal space_ $(V,Q)$ of dimension $m$ is

$\displaystyle\tau(V)=\begin{cases}1&\text{if }m\text{ is odd, }q\equiv
1\allowbreak\mkern 10.0mu({\operator@font mod}\,\,4),Q\sim\sum x_{i}^{2},\\\
-1&\text{if }m\text{ is odd, }q\equiv 1\allowbreak\mkern
10.0mu({\operator@font mod}\,\,4),Q\sim b\sum x_{i}^{2},\\\ \iota^{m}&\text{if
}m\text{ is odd, }q\equiv 3\allowbreak\mkern 10.0mu({\operator@font
mod}\,\,4),Q\sim\sum x_{i}^{2},\\\ (-\iota)^{m}&\text{if }m\text{ is odd,
}q\equiv 3\allowbreak\mkern 10.0mu({\operator@font mod}\,\,4),Q\sim b\sum
x_{i}^{2},\end{cases}$

where $\iota\in\mathbb{C}$ satisfies $\iota^{2}=-1$, and
$b\in\mathbb{F}_{q}\setminus\mathbb{F}_{q}^{2}$. More generally, when $V$ is
orthogonal direct sum $V_{1}\oplus V_{2}\oplus\cdots\oplus V_{l}$, the type is
defined by $\tau(V)=\prod\limits_{i=1}^{l}\tau(V_{i})$.

Now we describe the conjugacy classes and the centralizer of an element in
finite symplectic and orthogonal groups. This is the classic work of [36].We
recall briefly the results therein, which will be used further. A symplectic
signed partition is a partition of a number $k$, such that the odd parts have
even multiplicity and even parts have a sign associated with it. The set of
all symplectic signed partitions will be denoted as $\Lambda_{\mathrm{Sp}}$.
An orthogonal signed partition is a partition of a number $k$, such that all
even parts have even multiplicity, and all odd parts have a sign associated
with it. The set of all orthogonal signed partitions will be denoted as
$\Lambda_{\mathrm{O}}$. The dual of a monic degree $r$ polynomial $f(x)\in
k[x]$ satisfying $f(0)\neq 0$, is the polynomial given by
$f^{*}(x)=f(0)^{-1}x^{r}f(x^{-1})$.

According to [36], [32], the conjugacy classes of $\mathrm{Sp}_{2n}(q)$ are
parameterized by the functions
$\lambda:\Phi\rightarrow\Lambda^{2n}\cup\Lambda_{\mathrm{Sp}}^{2n}$, where
$\Phi$ denotes the set of all monic, non-constant, irreducible polynomials,
$\Lambda^{2n}$ is the set of all partitions of $1\leq k\leq 2n$ and
$\Lambda_{\mathrm{Sp}}^{2n}$ is the set of all symplectic signed partitions of
$1\leq k\leq 2n$. Such a $\lambda$ represent a conjugacy class of
$\mathrm{Sp}$ if and only if (a) $\lambda(u)=0$, (b)
$\lambda_{\varphi^{*}}=\lambda_{\varphi}$, (c)
$\lambda_{\varphi}\in\Lambda^{n}_{\mathrm{Sp}}$ iff $\varphi=u\pm 1$ (we
distinguish this $\lambda$, by denoting it $\lambda^{\pm}$) and, (d)
$\displaystyle\sum_{\varphi}|\lambda_{\varphi}|\textup{deg}(\varphi)=2n$. Also
from [36], [32], we find that a similar kind of statement is true for the
groups $\mathrm{O}^{\epsilon}_{n}(q)$. The conjugacy classes of
$\mathrm{O}^{\epsilon}_{n}(q)$ are parameterized by the functions
$\lambda:\Phi\rightarrow\Lambda^{n}\cup\Lambda_{\mathrm{O}{}}^{n}$, where
$\Phi$ denotes the set of all monic, non-constant, irreducible polynomials,
$\Lambda^{n}$ is the set of all partitions of $1\leq k\leq n$ and
$\Lambda_{\mathrm{O}}^{n}$ is the set of all symplectic signed partitions of
$1\leq k\leq n$. Such a $\lambda$ represent a conjugacy class of
$\mathrm{O}^{\epsilon}_{n}(q)$ for $\epsilon=\pm$, if and only if (a)
$\lambda(x)=0$, (b) $\lambda_{\varphi^{*}}=\lambda_{\varphi}$, (c)
$\lambda_{\varphi}\in\Lambda^{n}_{\mathrm{O}}$ iff $\varphi=u\pm 1$ (we
distinguish this $\lambda$, by denoting it $\lambda^{\pm}$) and, (d)
$\displaystyle\sum_{\varphi}|\lambda_{\varphi}|\textup{deg}(\varphi)=n$. Class
representative corresponding to given data can be found in [34], [33], [10]
and we will mention them whenever needed. We mention the following results
about the conjugacy class size (and hence the size of the centraliser) of
elements corresponding to given data, which can be found in [36].

######  Lemma 2.1.

[36, pp. 36] Let $X\in\mathrm{Sp}_{2n}(q)$ be a matrix corresponding to the
data $\Delta(X)=\\{(\phi,\mu_{\phi}):\phi\in\Phi_{X}\subset\Phi\\}$. Then the
conjugacy class of $X$ in $\mathrm{Sp}_{2n}(q)$ is of size
$\dfrac{|\mathrm{Sp}_{2n}(q)|}{\prod\limits_{\phi}B(\phi)}$ where $B(\phi)$
and $A(\phi^{\mu})$ are defined as follows

$\displaystyle A(\phi^{\mu})=\begin{cases}|\mathrm{U}(m_{\mu},Q)|&\text{if
}\phi(x)=\phi^{*}(x)\neq x\pm 1\\\
|\mathrm{GL}(m_{\mu},Q)|^{\frac{1}{2}}&\text{if }\phi\neq\phi^{*}\\\
|\mathrm{Sp}(m_{\mu},q)|&\text{if }\phi(x)=x\pm 1,~{}\mu\text{ odd}\\\
|q^{\frac{1}{2}m_{\mu}}\mathrm{O}^{\epsilon}(m_{\mu},q)|&\text{if
}\phi(x)=x\pm 1,~{}\mu\text{ even}\end{cases},$

where $\epsilon$ gets determined by the sign of the corresponding partition,
$Q=q^{|\phi|}$, $m_{\mu}=m(\phi^{\mu})$ and

$\displaystyle B(\phi)=Q^{\sum\limits_{\mu<\nu}\mu
m_{\mu}m_{\nu}+\frac{1}{2}\sum\limits_{\mu}(\mu-1)m_{\mu}^{2}}\prod\limits_{\mu}A(\phi^{\mu}).$

######  Lemma 2.2.

[36, pp. 39] Let $X\in\mathrm{O}^{\epsilon}_{n}(q)$ be a matrix corresponding
to the data $\Delta(X)=\\{(\phi,\mu_{\phi}):\phi\in\Phi_{X}\subset\Phi\\}$.
Then the conjugacy class of $X$ in $\mathrm{O}^{\epsilon}(q)$ is of size
$\dfrac{|\mathrm{Sp}|}{\prod\limits_{\phi}B(\phi)}$ where $B(\phi)$ and
$A(\phi^{\mu})$ are defined as before, except when $\phi(x)=x\pm 1$,

$\displaystyle
A(\phi^{\mu})=\begin{cases}|\mathrm{O}^{\epsilon^{\prime}}(m_{\mu},q)|&\text{if
}\mu\text{ odd}\\\ q^{-\frac{1}{2}m_{\mu}}|\mathrm{Sp}(m_{\mu},q)|&\text{if
}\mu\text{ even}\end{cases},$

where $\epsilon^{\prime}$ in $\mathrm{O}{\epsilon^{\prime}}_{m_{\mu}}(q)$ gets
determined by the corresponding sign of the part, of the partition.

The quantity $B(\phi)$ will also be denoted as
$c_{\mathrm{Sp},\phi,q}(\lambda^{\pm})$ in case of symplectic groups and as
$c_{\mathrm{O}^{\epsilon},\phi,q}(\lambda^{\pm})$ in case of orthogonal groups
when $\phi=x\pm 1$.

###  2.3. Unitary groups

Consider the field $k=\mathbb{F}_{q^{2}}$, and the involution
$\sigma\colon\mathbb{F}_{q^{2}}\longrightarrow\mathbb{F}_{q^{2}}$ defined as
$\sigma(a):=\overline{a}=a^{q}$ of the field. We further get an automorphism
of $\mathbb{F}_{q^{2}}[t]$, the polynomial ring over $\mathbb{F}_{q^{2}}$, by
action on the coefficients of the polynomials. The image of
$f\in\mathbb{F}_{q^{2}}[t]$ will be denoted as $\overline{f}$. Consider the
Hermitian form given by the matrix

$\Pi_{n}=\begin{pmatrix}0&0&\ldots&0&1\\\ 0&0&\ldots&1&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&1&\ldots&0&0\\\
1&0&\ldots&0&0\end{pmatrix}.$

Then we identify the unitary group (corresponding to the above Hermitian form)
with the set

$\mathrm{U}(n,\mathbb{F}_{q^{2}})=\left\\{A\in\mathrm{GL}(n,\mathbb{F}_{q^{2}})\mid
A\Pi_{n}\overline{A}^{t}=\Lambda_{n}\right\\}.$

Since all non-degenerate Hermitian forms on an $n$-dimensional vector space
are unitarily congruent to the Hermitian form given by the above matrix, it is
evident that all the unitary groups arising are conjugate with each other
inside $\mathrm{GL}_{n}(q^{2})$ (see theorem $10.3$ of [12]). From the work of
Wall, it is known that two elements $g_{1},g_{2}\in\mathrm{U}_{n}(q^{2})$ are
conjugate to each other if and only if they are conjugate in
$\mathrm{GL}_{n}(q^{2})$. These conjugacy classes are parameterized by special
polynomials and partitions. We call a polynomial $f$ of degree $d$ to be
_$\sim$ -symmetric_ if $\widetilde{f}=f$ where
$\widetilde{f}(t)=\overline{f(0)}^{-1}t^{d}\bar{f}(t^{-1})$. A polynomial
without a proper $\sim$-symmetric factor will be called a _$\sim$
-irrdeucible_ polynomial. Note that a $\sim$-irreducible polynomial can be
reducible in the usual sense. Denote by $\widetilde{\Phi}$ to be the set of
all $\sim$-irreducible polynomials. Let $\mathcal{P}_{n}$ denote the set of
all partitions of numbers $\leq n$. Then a conjugacy class in
$\mathrm{U}(n,\mathbb{F}_{q^{2}})$ is determined by a function
$\lambda\colon\widetilde{\Phi}\longrightarrow\mathcal{P}_{n}$ which satisfies
(a) $\lambda(f)=\lambda(\widetilde{f})$ and, (b)
$\sum\limits_{\phi}|\lambda(\phi)|\deg\phi=n$. The centralizer six=ze can be
found in [36] and we are not quoting it here.

Using these information, we determine when a matrix $A$ is an $M$-th root of
identity. This information is further used in subsequent sections to determine
the desired generating functions, with the help of the concept of central join
of two matrices, following [34], [33], [35].

##  3\. Semisimple & unipotent roots of identity

###  3.1. Semisimple classes

Note that an element in $\mathrm{G}(q)$ is semisimple if and only if it has
order coprime to $q$. In this subsection, we classify all the semisimple
elements, which give $\mathrm{Id}$, when raised to the power $M$. In this
subsection, we will assume that $(M,q)=1$. Let $f$ denote a product of
distinct irreducible polynomials, and $A$ be the semisimple matrix with $f$ as
its characteristic polynomial. Then if $A^{M}=\mathrm{Id}$, then $f$ must have
irreducible factors coming from irreducible factors of
$x^{M}-1\in\mathbb{F}_{q}[x]$. We recall some results and definitions from
[21]. For a positive integer $n$, the splitting field of $x^{n}-1$ over a
field $K$ is called the $n$-th _cyclotomic field_ over $K$ and denoted by
$K^{(n)}$. The roots of $x^{n}-1$ in $K^{(n)}$ are called the $n$-th roots of
unity over $K$ and the set of all these roots is denoted by $E^{(n)}$.

######  Lemma 3.1.

[21, Theorem 2.42] Let $n$ be a positive integer and $K$ a field of
characteristic $p$. If $(p,n)=1$, then $E^{(n)}$ is a cyclic group of order
$n$ with respect to multiplication in $K^{(n)}$.

A generator of the cyclic group is called _primitive $n$-th root of unity over
$K$_. If $(n,p)=1$ and $\zeta$ is a primitive $n$-th root of unity over $K$,
the polynomial

$\displaystyle Q_{n}(x)=\prod\limits_{\begin{subarray}{c}s=1\\\
(s,n)=1\end{subarray}}^{n}\left(x-\zeta^{s}\right),$

is called the _$n$ -th cyclotomic polynomial_ over $K$. When $(n,p)=1$, we
have that $x^{n}-1=\prod_{d|n}Q_{d}(x)$, see [21, Theorem 2.45]. Note that if
$A^{M}=1$, for each irreducible factor $g$ of $f$, there exists an $m(g)$ such
that $f|Q_{m(g)}(x)$. Thus we need to know how $Q_{n}(x)$ factors over
$\mathbb{F}_{q}[x]$. We recall the following result.

######  Lemma 3.2.

[21, Theorem 2.47(ii)] When $K=\mathbb{F}_{q}$ and $(n,q)=1$, then the
cyclotomic polynomial $Q_{n}$ factors into $\phi(n)/e(n)$ distinct monic
irreducible polynomials in $K[x]$ of the same degree $e(n)$, where $e(n)$ is
order of $q$ in $\mathbb{Z}/n\mathbb{Z}^{\times}$.

Since $(M,p)=1$, the polynomial $x^{M}-1$ is separable and hence for $d\neq
d^{\prime}$ dividing $M$, the cyclotomic polynomials $Q_{d}(x)$ and
$Q_{d^{\prime}}(x)$ are coprime. Hence we have the following lemma.

######  Proposition 3.3.

Let $(M,q)=1$ and $\mathrm{G}_{n}=\mathrm{GL}_{n}(q)$. Let $a_{n}$ denote the
number of semisimple conjugacy classes $x_{n,i}^{G}$, such that
$x_{n,i}^{M}=\mathrm{Id}_{n}$ for some indices $i$. Then we have that

(3.1)
$1+\sum\limits_{n=1}^{\infty}a_{n}z^{n}=\prod\limits_{\begin{subarray}{c}d|M\end{subarray}}\left(\dfrac{1}{1-z^{e(d)}}\right)^{\frac{\phi(d)}{e(d)}},$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$.

###### Proof.

Let $A$ be a semisimple matrix satisfying $A^{M}=1$. Then the characteristic
polynomial of $A$, divides $x^{M}-1$. Since $(M,q)=1$, using factorization of
$x^{M}-1$ and Lemma 3.2, we get that the irreducible factors of $x^{M}-1$ are
of degree $\phi(d)/e(d)$ for $d|M$, where $e(d)$ denotes the multiplicative
order of $q$ in $\mathbb{Z}/d\mathbb{Z}^{\times}$. Furthermore, $(M,q)=1$
implies that the polynomial $x^{M}-1$ is separable. Hence for $d\neq
d^{\prime}$ dividing $M$, we have that $(Q_{d}(x),Q_{d^{\prime}}(x))=1$. Hence
the result follows. ∎

To deduce the result on the probability of a semisimple element to be an
$M$-th root of identity, we need to know the order of the centralizer of each
element. This has been discussed in Section 2.

######  Corollary 3.4.

Let $(M,q)=1$ and $\mathrm{G}_{n}=\mathrm{GL}_{n}(q)$. Let $b_{n}$ denote the
proportion of semisimple $M$-th roots of identity in $\mathrm{G}_{n}$. Then we
have that

$1+\sum\limits_{n=1}^{\infty}b_{n}z^{n}=\prod\limits_{\begin{subarray}{c}d|M\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{me(d)}}{|\mathrm{GL}_{m}(q^{e(d)})|}\right)^{\frac{\phi(d)}{e(d)}},$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$.

To calculate the number of semisimple conjugacy classes in
$\mathrm{Sp}_{2n}(q)$, which are $M$-th root of $\mathrm{Id}$, we need to know
the factorization of $x^{M}-1$. Before proceeding further, for an integer $m$,
let us define

$\displaystyle D_{m}=\\{x\in\mathbb{Z}_{>0}:x|q^{m}+1,x\nmid q^{t}+1,\text{
for all }0\leq t<m\\}.$

The set $D_{m}$ plays a crucial role in counting the semisimple elements in
finite symplectic and orthogonal groups, which gives $\mathrm{Id}$ on raising
the power to $M$. We collect the following results from [37].

######  Lemma 3.5.

[37, Theorem 8, and Theorem 11] We have the following results.

  1. (1)

An irreducible polynomial of degree $2m$ over $\mathbb{F}_{q}$ is SRIM iff
$\mathrm{ord}(f)\in D_{m}$

  2. (2)

For $d\in D_{m}$ the $d$-th cyclotomic polynomial $Q_{d}$ factors into product
of all distinct SRIM polynomials of degree $2m$ and order $d$.

######  Corollary 3.6.

The polynomial $Q_{d}(x)$ has a (and hence all) $*$-symmetric irreducible
factor if and only if $d\in D_{m}$ for some $m$. Furthermore, if $e(d)$ is
odd, $Q_{d}$ has no SRIM factor.

###### Proof.

By definition, $Q_{n}$ is a $*$-symmetric polynomial. Hence its factors are
either irreducible $*$-symmetric or product of the form $gg^{*}$, where $g\neq
g^{*}$ and $g$ are irreducible. If $e(n)$ is odd, then there is no irreducible
$*$-symmetric factor of $Q_{n}$, since all $*$-symmetric irreducible
polynomials are of even degree. So we assume that $e(n)$ is even. So assume
$e(n)$ is even. The backward direction is the content of Lemma 3.5. In the
other direction, assume that $Q_{n}(x)$ has a $*$-symmetric polynomial, say
$f$. Then $\mathrm{ord}(f)\in D_{m}$ for some $m$. But, by definition of
$Q_{n}$, we have that $\mathrm{ord}(f)=n$ and hence $n\in D_{m}$. ∎

######  Proposition 3.7.

Let $(M,q)=1$ and $G=\mathrm{Sp}_{2n}(q)$. Let $a_{n}$ denote the number of
semisimple conjugacy classes $x_{n,i}^{G}$, such that
$x_{n,i}^{M}=\mathrm{Id}_{n}$ for some indices $i$. Then we have that

(3.2)
$1+\sum\limits_{n=1}^{\infty}a_{n}z^{n}=\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in
D_{m}\end{subarray}}\left(\dfrac{1}{1-z^{e(d)}}\right)^{\frac{\phi(d)}{e(d)}}\prod\limits_{\begin{subarray}{c}d|M\\\
d\not\in
D_{m}\end{subarray}}\left(\dfrac{1}{1-z^{2e(d)}}\right)^{\frac{\phi(d)}{2e(d)}}\right)$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$.

###### Proof.

Let $A\in\mathrm{Sp}_{2n}(q)$ be a matrix satisfying $A^{M}=\mathrm{Id}$. Then
we have that, the minimal polynomial of $A$ over $\mathbb{F}_{q}$ should
divide $x^{M}-1$. Since $x^{M}-1=\prod\limits_{d|M}Q_{d}(x)$ and $Q_{d}(x)$
further decomposes into irreducibles we need to do a case-by-case study.
Because of Corollary 3.6 we have to use $D_{m}$’s. So first assume that $d\in
D_{m}$, then each of the factors of $Q_{d}$ determine a Symplectic matrix of
$\mathrm{Sp}_{e(d)}$ which gives $\mathrm{Id}$ when raised to the power $M$.
Since $Q_{d}$ has in total $\phi(d)/e(d)$ many irreducible factors, this
justifies the first factor in the product.

Next, suppose $d\not\in D_{m}$. Then for any $g\neq g^{*}$, a factor of
$Q_{d}$, we need to pair up $g$ and $g^{*}$ to get a conjugacy class of
$\mathrm{Sp}_{2e(d)}$, which contributes to the counting. Since $Q_{d}$ is
$*$-symmetric, $g$ and $g^{*}$ occur together in the factorization of $Q_{d}$.
Furthermore, $Q_{d}$ is separable and hence all the irreducible factors are
coprime to each other. Since $g$ and $g^{*}$ are combined together, the
exponent of $z$ is taken to be $2e(d)$. But in the process, we are grouping
$g$ and $g^{*}$ together, resulting in the power of the factor being
$\phi(d)/2e(d)$. This finishes the proof. ∎

######  Corollary 3.8.

Let $(M,q)=1$ and $\mathrm{G}_{n}=\mathrm{Sp}_{2n}(q)$. Let $b_{n}$ denote the
proportion of semisimple $M$-th roots of identity in $\mathrm{G}_{n}$. Then we
have that

$\displaystyle\begin{split}1+\sum\limits_{n=1}^{\infty}b_{n}z^{n}&=\left(1+\sum\limits_{m=1}^{\infty}\dfrac{u^{m}}{|\mathrm{Sp}_{2m}(q)|}\right)^{o(M)}\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in
D_{m}\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{me(d)}}{|\mathrm{U}_{m}(q^{2e(d)})|}\right)^{\frac{\phi(d)}{e(d)}}\right.\\\
&\times\left.\prod\limits_{\begin{subarray}{c}d|M\\\ d\not\in D_{m}\\\ d\neq
1,2\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{me(d)}}{|\mathrm{GL}_{m}(q^{e(d)})|}\right)^{\frac{\phi(d)}{2e(d)}}\right)\end{split}$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$, and $o(M)=2$ if $2|M$ and $1$ otherwise.

###### Proof.

Start with the case when $M$ is even. In this case, both $\mathrm{Id}$ and
$\mathrm{Id}$ are $M$-th roots of identity. Now let us assume that $A$ is an
$M$-th root of identity, such that $\pm 1$ is not an eigenvalue of $A$. Then
the characteristic polynomial of $A$ is a product of factors of $Q_{d}$, where
$d|M$ and the polynomials are irreducible of degree more than $1$. The rest of
the proof follows from the description of the centralizers when the
irreducible factor is a SRIM polynomial or not. In the latter case, we have
that $g\neq g^{*}$ occurs together in the factorization of the characteristic
polynomial of $A$. For $M$ being even $-\mathrm{Id}$ is not an $M$-th root of
identity, which proves the existence of the function $o(M)$ in the formula.
This finishes the proof. ∎

######  Corollary 3.9.

Let $(M,q)=1$ and $\mathrm{G}_{2n}=\mathrm{O}^{\epsilon}_{2n}(q)$ and
$\mathrm{G}_{2n+1}=\mathrm{O}_{2n+1}(q)$, where $\epsilon=\pm$. Let
$b_{n}^{\epsilon}$ and $b_{n}$ denote the proportion of semisimple $M$-th
roots of identity in $\mathrm{G}_{2n}$ and $\mathrm{G}_{2n+1}$ respectively.
Let us define

$\displaystyle
B_{+}(z)=1+\sum\limits_{n=1}^{\infty}b_{n}^{+}z^{n},B_{-}(z)=\sum\limits_{n=1}^{\infty}b_{n}^{-}z^{n},B(z)=1+\sum\limits_{n=1}^{\infty}b_{n}z^{n}.$

Then we have that

$\displaystyle B_{+}(z^{2})+B_{-}(z^{2})+2zB(z^{2})$ $\displaystyle=$
$\displaystyle\left(1+\sum\limits_{m=1}^{\infty}\left(\dfrac{1}{|\mathrm{O}^{+}_{2m}(q)|}+\dfrac{1}{|\mathrm{O}^{-}_{2m}(q)|}\right)z^{2m}+z\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{2m}}{|\mathrm{Sp}_{2m}(q)|}\right)\right)^{o(M)}$
$\displaystyle\times$
$\displaystyle\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in
D_{m}\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{2me(d)}}{|\mathrm{U}_{m}(q^{2e(d)})|}\right)^{\frac{\phi(d)}{e(d)}}\prod\limits_{\begin{subarray}{c}d|M\\\
d\not\in D_{m}\\\ d\neq
1\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{2me(d)}}{|\mathrm{GL}_{m}(q^{e(d)})|}\right)^{\frac{\phi(d)}{2e(d)}}\right)$

and

$\displaystyle B_{+}(z^{2})-B_{-}(z^{2})$ $\displaystyle=$
$\displaystyle\left(1+\sum\limits_{m=1}^{\infty}\left(\dfrac{1}{|\mathrm{O}^{+}_{2m}(q)|}-\dfrac{1}{|\mathrm{O}^{-}_{2m}(q)|}\right)z^{2m}\right)^{o(M)}$
$\displaystyle\times$
$\displaystyle\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in
D_{m}\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{(-1)^{m}z^{2me(d)}}{|\mathrm{U}_{m}(q^{2e(d)})|}\right)^{\frac{\phi(d)}{e(d)}}\prod\limits_{\begin{subarray}{c}d|M\\\
d\not\in D_{m}\\\ d\neq
1\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{2me(d)}}{|\mathrm{GL}_{m}(q^{e(d)})|}\right)^{\frac{\phi(d)}{2e(d)}}\right)$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$, and $o(M)=2$ if $2|M$ and $1$ otherwise.

###### Proof.

The proof is almost similar to Corollary 3.8, but care needs to be taken as we
are giving it in a combinatorial formula. This formula occurs because of the
existence of different quadratic forms. Let $X$ be an orthogonal $M$-th root
of unity, preserving a non-degenerate quadratic form on an $N$ dimensional
vector space $V$, where $N$ is even or odd. Consider the characteristic
polynomial of $X$, which decomposes into irreducibles with factors being
$\varphi=\varphi^{*}$ and irreducible or $\psi\psi^{*}$, with
$\psi\neq\psi^{*}$ irreducible. We use the centralizer information from
Section 2.

Let us start with the first equation. The characteristic polynomial of $X$
might have root to be $1$ or $-1$. But, if $M$ is odd, it will not have $-1$
as a root. This justifies the presence of $o(M)$. Note that for $N$ being even
and positive the coefficient of $z^{N}$ is sum of the probabilities in case of
$\mathrm{O}^{+}_{N}(q)$ and $\mathrm{O}^{-}_{N}(q)$. Furthermore for odd $N$
this is two times the probability in the case of $\mathrm{O}_{N}(q)$, which
occurs because of two quadratic forms. In case of odd $N$, note that
$|\mathrm{O}_{N}(q)|=2|\mathrm{Sp}_{N-1}(q)|$, which we are using in the
formula. This proves the equality in the first equation.

We prove the next equation by modifying the first equation. We need the value
of type of an orthogonal space, as was discussed in Section 2. This
consideration doesn’t alter the first factor, as they correspond to the case
whether $\mathrm{Id}$ and $-\mathrm{Id}$ are $M$-th roots of identity or not.
In the right side of the first equation, for the self conjugate irreducible
factors, replace $z$ vy $\pm iz$, according to their types. This proves the
second equality. ∎

Let us now deduce the SCIM factors of $x^{M}-1$ in $\mathbb{F}_{q^{2}}$.
According to [5, Lemma 2], if $f$ is a SCIM polynomial of odd degree $d$, then
for any root $\zeta$ of $f$, it satisfies $\zeta^{q^{d}+1}=1$. This means that
$\mathrm{ord}(f)|q^{d}+1$. For a positive integer $m$ define

$\displaystyle\widetilde{D}_{m}=\\{y\in\mathbb{Z}_{>0}:y|q^{2m+1}+1,y\nmid
q^{t}+1\text{ for odd }1\leq t<2m+1\\}.$

It is evident that if $f$ is a SCIM polynomial of degree $d$, then
$\mathrm{ord}(f)\in\widetilde{D}_{(d-1)/2}$. If $d\in D_{m}$, and $\beta$ is a
primitive $d$-th root of unity, then the set
$\\{\beta,\beta^{q^{2}},\beta^{q^{4}},\cdots,\beta^{q^{2(d-1)}}\\}$ is of
cardinality $d$. Indeed, if $\beta^{q^{2i}}=\beta^{q^{2j}}$ for some $0\leq
i<j\leq d-1$, then $d|q^{2(j-i)}-1$. Hence $2(j-i)$ is an even multiple of
$d$, which forces $2(j-i)=0$, see [37, Prposition 1]. For
$e\in\widetilde{D}_{m}$ and a primitive $e$-th root of unity, let us define

$\displaystyle
f_{\beta}(x)=\prod\limits_{i=0}^{d-1}\left(x-\beta^{q^{2i}}\right).$

Then by [22, Theorem 3.4.8] this is an irreducible polynomial. We claim that
$f_{\beta}$ is a SCIM polynomial. Note that
$\left(\beta^{q^{2j}}\right)^{-q}=\beta^{q^{2(m+i+1)}}$, since
$\beta^{q^{2m+1}+1}=1$. This implies that $\left(\beta^{q^{2j}}\right)^{-q}$
is also a root, and hence $f_{\beta}$ is a SCIM polynomial, see [9, pp. 23].
Using [21, Theorem 2.47], we conclude that for an irreducible polynomial $f$
of degree $2d+1$ over $\mathbb{F}_{q^{2}}$, the following statements are
equivalent;

  1. (1)

$f$ is SCIM,

  2. (2)

$\mathrm{ord}(f)\in\widetilde{D}_{d}$,

  3. (3)

$f=\widetilde{f}_{\beta}$ for some $\beta$ being a $t$-th root of unity for
some $t\in\widetilde{D}_{d}$.

For $t\in\widetilde{D}_{d}$, we have that $Q_{d}(x)=\prod(x-\gamma)$, where
$\gamma$ runs over all $t$-th primitive root of unity. It follows that in this
case $Q_{d}$ factors into a product of all SCIM polynomials. Using a similar
argument as in Corollary 3.6 we conclude the following.

######  Lemma 3.10.

The polynomial $Q_{d}(x)$ has a (and hence all) $\sim$-symmetric irreducible
factor if and only if $d\in\widetilde{D}_{m}$ for some $m$. Furthermore, if
$e(d)$ is even, $Q_{d}$ has no SCIM factor.

######  Proposition 3.11.

Let $(M,q)=1$ and $G=\mathrm{U}_{n}(q^{2})$. Let $a_{n}$ denote the number of
semisimple conjugacy classes $x_{n,i}^{G}$, such that
$x_{n,i}^{M}=\mathrm{Id}_{n}$ for some indices $i$. Then we have that

(3.3)
$1+\sum\limits_{n=1}^{\infty}a_{n}z^{n}=\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in\widetilde{D}_{m}\end{subarray}}\left(\dfrac{1}{1-z^{e(d)}}\right)^{\frac{\phi(d)}{e(d)}}\prod\limits_{\begin{subarray}{c}d|M\\\
d\not\in\widetilde{D}_{m}\end{subarray}}\left(\dfrac{1}{1-z^{2e(d)}}\right)^{\frac{\phi(d)}{2e(d)}}\right)$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$.

###### Proof.

Let $A\in\mathrm{U}_{n}(q^{2})$ be a matrix satisfying $A^{M}=\mathrm{Id}$.
Then we have that, the minimal polynomial of $A$ over $\mathbb{F}_{q^{2}}$
should divide $x^{M}-1$. Since $x^{M}-1=\prod\limits_{d|M}Q_{d}(x)$ and
$Q_{d}(x)$ further decomposes into irreducibles we examine this each $d$ case-
by-case depending on whether $d$ is inside or outside $\widetilde{D}_{m}$ for
some $m$, and use Lemma 3.10. So first assume that $d\in\widetilde{D}_{m}$,
then each of the factors of $Q_{d}$ determine a Unitary matrix of
$\mathrm{U}_{e(d)}(q^{2})$ which gives $\mathrm{Id}$ when raised to the power
$M$. Since $Q_{d}$ has in total $\phi(d)/e(d)$ many irreducible factors, this
justifies the first factor in the product.

Next, suppose $d\not\in D_{m}$. Then for any $g\neq\widetilde{g}$, a factor of
$Q_{d}$, we need to pair up $g$ and $g^{*}$ to get a conjugacy class of
$\mathrm{U}_{e(d)}(q^{2})$, which contributes to the counting. Since $Q_{d}$
is $\sim$-symmetric (because $\alpha$ and $\alpha^{-q}$ are roots of $Q_{d}$,
as $(d,M)=1$), $g$ and $\widetilde{g}$ occur together in the factorization of
$Q_{d}$. Furthermore, $Q_{d}$ is separable and hence all the irreducible
factors are coprime to each other. Hence the exponent of $z$ is taken to be
$2e(d)$. But in the process, we are grouping $g$ and $g^{*}$, resulting in the
power of the factor being $\phi(d)/2e(d)$. This finishes the proof. ∎

######  Corollary 3.12.

Let $(M,q)=1$ and $\mathrm{G}_{n}=\mathrm{U}_{n}(q^{2})$. Let $b_{n}$ denote
the proportion of semisimple $M$-th roots of identity in $\mathrm{G}_{n}$.
Then we have $1+\sum\limits_{n=1}^{\infty}b_{n}z^{n}$ to be equal to

(3.4)
$\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in\widetilde{D}_{m}\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{me(d)}}{|\mathrm{U}_{m}(q^{2e(d)})|}\right)^{\frac{\phi(d)}{e(d)}}\prod\limits_{\begin{subarray}{c}d|M\\\
d\not\in\widetilde{D}_{m}\end{subarray}}\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{2me(d)}}{|\mathrm{GL}_{m}(q^{2e(d)})|}\right)^{\frac{\phi(d)}{2e(d)}}\right)$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$.

###### Proof.

This follows easily from the information on centralizers and size of the
conjugacy classes. The flow of reasoning is as same as Corollary 3.4,
Corollary 3.8 and Corollary 3.9. Hence it is left to the reader. ∎

###  3.2. Unipotent classes

Recall that an element $x\in G(q)$ is a unipotent element if and only if
$\mathrm{ord}(x)$ is a power of the defining prime. Hence to find all
unipotent elements of order $M$, it is enough to find elements of order
$p^{r}$ where $M=t\cdot p^{r}$ and $p\nmid t$. In the following result, we
find the conjugacy classes of unipotent elements of order $p^{r}$ when
$G=\mathrm{GL}_{n}$. In the other finite groups of Lie type two unipotent
elements $u_{1}$ and $u_{2}$ need not be $G$-conjugate, even if they are
$\mathrm{GL}_{n}$ conjugate. But it so happens that the unipotent conjugacy
class in other Finite groups of Lie type descend from those in
$\mathrm{GL}_{n}$. Thus it is necessary that we first resolve the case of
$\mathrm{GL}_{n}(q)$. The treatments for the other groups will follow easily.

######  Proposition 3.13.

Let $\lambda=\lambda_{1}^{i_{1}}\lambda_{2}^{i_{2}}\ldots\lambda_{k}^{i_{k}}$
be a partition of $|\lambda|$, where
$\lambda_{1}>\lambda_{2}>\ldots>\lambda_{k}$. Consider $J_{\lambda}$ to be the
matrix corresponding to the Jordan canonical form of the unipotent matrix
attached with $\lambda$. Then $J_{\lambda}$ is of order $p^{r}$ if and only if
$p^{r-1}<\lambda_{1}\leq p^{r}$.

###### Proof.

Note that $N_{\lambda}=J_{\lambda}-\mathrm{Id}=(a_{ij})$ is a nilpotent matrix
with the $(i,i+1)$-th entry $a_{ii+1}\in\\{0,1\\}$ for all $1\leq
i\leq|\lambda|-1$ and $a_{ij}=0$ for all $(i,j)\neq(i,i+1)$. Also
$J_{\lambda}-\mathrm{Id}$ is a nilpotent matrix satisfies
$(J_{\lambda}-\mathrm{Id})^{|\lambda|}=0$. Given that
$J_{\lambda}\in\mathrm{GL}_{|\lambda|}(\mathbb{F}{p})$ is a nontrivial
unipotent element, it must have order $p^{t}$ for some $t>0$. Using the
binomial theorem,

$\displaystyle
J_{\lambda}^{p^{t}}=\mathrm{Id}+\displaystyle\sum\limits_{i=1}^{p^{t}}{p^{t}\choose
i}N_{\lambda}^{i}.$

Since $\lambda=\ell_{1}^{i_{1}}\ell_{2}^{i_{2}}\ldots\ell_{k}^{i_{k}}$, the
$t$ should satisfy $p^{t}\geq\ell_{1}$. This proves both sides of the
statement. ∎

######  Lemma 3.14.

Let $Q(r,[s])$ denote the number of partitions of $r$, with parts not
exceeding $s$. Then the generating function for $Q$ in indeterminate $z$ is
given by

$\displaystyle
1+\sum\limits_{r=1}^{\infty}Q(r,[s])z^{r}=\prod\limits_{t=1}^{s}\dfrac{1}{1-z^{t}}.$

##  4\. Generating functions for all roots of identity

To detect an element $x\in\mathrm{G}(q)$, such that $x^{M}=\mathrm{Id}$, we
use the Jordan decomposition. First, let us write $M=t\cdot p^{r}$, where
$t\nmid p$. Using Jordan decomposition (see [24]), we write $x=x_{s}x_{u}$,
where $x_{s}$ is semisimple and $x_{u}$ is unipotent. It can be easily seen
that $x^{M}=\mathrm{Id}$ if and only if $x_{s}^{t}=\mathrm{Id}$ and
$x_{u}^{p^{r}}=\mathrm{Id}$. Since we use the combinatorial data to conclude
our results, it is important to know the unipotent part of $x$, when $x$ is
determined by a combinatorial data $\Delta(x)$. For any element
$x\in\mathrm{G}(q)$, we consider it inside ambient $\mathrm{GL}_{n}(q)$, and
since the order of an element is unaltered undeithis consideration, we discuss
the case for an element in $\mathrm{GL}_{n}$ only. The other cases will follow
easily. Recall from Section 2, for an element $x\in\mathrm{GL}_{n}(q)$, if the
combinatorial data is given by
$\\{(f,\lambda_{f}(x)):f\in\Phi\setminus\\{z\\}\\}$, then the companion matrix
corresponding to the combinatorial data is given by the sum of the blocks of
the form

$\displaystyle\begin{pmatrix}C_{f}&\mathrm{Id}&&\\\ &C_{f}&&\\\ &&\ddots&\\\
&&&C_{f}\end{pmatrix},$

where $C_{f}$ is the companion matrix corresponding to $f$, and $\mathrm{Id}$
is a $\deg f\times\deg f$ matrix. Also, the size of this blocks are
$\lambda_{i}\deg(f)\times\lambda_{i}\deg(f)$, where
$\lambda=(\lambda_{1},\lambda_{2},\ldots)$. It is well known that two matrices
$x,x^{\prime}\in\mathrm{GL}_{n}$ are conjugate to each other if and only if
they are conjugate in $\mathrm{GL}_{n}(\overline{\mathbb{F}_{q}})$, see [3].
Hence we consider the matrix $x$ in
$\mathrm{GL}_{n}(\overline{\mathbb{F}_{q}})$, up to conjugacy. Then the
combinatorial data $(f,\lambda_{f}(x))$ becomes
$\\{(x-\mu_{1},\lambda_{f}(x)),\ldots,(x-\mu_{s},\lambda_{f}(x))\\}$, where
$f(x)=(x-\mu_{1})\ldots(x-\mu_{s})$. Hence to summarize we have the following
result, using Proposition 3.13.

######  Lemma 4.1.

Let $\Delta(x)=\\{(f,\lambda_{f}(x)):f\in\Phi\\}$ be the combinatorial data
corresponding to $x$ and $M=t\cdot p^{r}$, $p\nmid t$. If $x=x_{s}x_{u}$ is
the Jordan decomposition of $x$, then $x$ is an $M$-th root of identity in
$\mathrm{GL}_{n}(q)$ if and only if $x_{s}$ is a $t$-th root of identity in
$\mathrm{GL}_{n}(q)$ and each part of $\lambda(f)$ has size less than or equal
to $p^{r}$.

In this section, we use cycle indices for classical groups to conclude the
final set of results. We will recall the cycle indices from [6] and [7]
whenever needed and then deduce our results. We assume the polynomials to be
in the indeterminate $u$. Let $x_{\varphi,\lambda}$ be variables corresponding
to pairs of polynomials and partitions. The cycle index for a group
$\mathrm{G}(q)$ is defined to be

$\displaystyle
1+\sum\limits_{n=1}^{\infty}\dfrac{z^{n}}{|\mathrm{G}_{n}(q)|}\left(\sum\limits_{\alpha\in\mathrm{G}_{n}(q)}\prod\limits_{\begin{subarray}{c}\varphi\neq
u\end{subarray}}x_{\varphi,\lambda_{\varphi}(\alpha)}\right).$

In the next couple of results, we are first going to recall the factorization
of the cycle indices, due to the works of Jason Fulman. After that, we are
going to substitute zero or one, on the basis of whether such an element
contributes to the quantity or not. We present the first lemma now.

######  Lemma 4.2.

[7, pp. 55] For $G(q)=\mathrm{GL}_{n}(q)$, we have

$\displaystyle
1+\sum\limits_{n=1}^{\infty}\dfrac{z^{n}}{|\mathrm{GL}_{n}(q)|}\left(\sum\limits_{\alpha\in\mathrm{GL}_{n}(q)}\prod\limits_{\begin{subarray}{c}\varphi\neq
u\end{subarray}}x_{\varphi,\lambda_{\varphi}(\alpha)}\right)$ $\displaystyle=$
$\displaystyle\prod\limits_{\varphi\neq u}\left(1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}x_{\varphi,\lambda}\dfrac{z^{n\cdot\deg\varphi}}{q^{\deg\varphi\cdot(\sum_{i}(\lambda_{i}^{\prime})^{2})}\prod\limits_{i\geq
1}\left(\dfrac{1}{q^{\deg\varphi}}\right)_{m_{i}(\lambda_{\varphi})}}\right).$

######  Theorem 4.3.

Let $a_{n}$ denote the number of elements in $\mathrm{GL}_{n}(q)$ which are
$M$-th root of unity. Let $M=t\cdot p^{r}$, where $p\nmid t$. Then the
generating function of the probability $a_{n}/|\mathrm{GL}_{n}(q)|$ is given
by

$\displaystyle
1+\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{|\mathrm{GL}_{n}(q)|}z^{n}=\prod\limits_{d|M}\left(1+\sum\limits_{m\geq
1}\sum\limits_{\begin{subarray}{c}\lambda\vdash m\\\ \lambda_{1}\leq
p^{r}\end{subarray}}\dfrac{z^{me(d)}}{q^{e(d)\cdot(\sum_{i}(\lambda_{i}^{\prime})^{2})}\prod\limits_{i\geq
1}\left(\dfrac{1}{q^{e(d)}}\right)_{m_{i}(\lambda_{\varphi})}}\right)^{\frac{\phi(d)}{e(d)}},$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$.

###### Proof.

It follows from the proof of Proposition 3.3 and Proposition 3.13 that, for an
element $x\in\mathrm{GL}_{n}(q)$,

  1. (1)

the semisimple part $x_{s}$ has order $t$ if and only if the irreducible
factors of the characteristic polynomial of $x_{s}$ divides $Q_{d}$ for some
$d|M$,

  2. (2)

the unipotent part $x_{u}$ has order $p^{r}$ if and only if the partition
corresponding to $x_{u}$ has all parts lesser than or equal to $p^{r}$.

Hence in the formula of Lemma 4.2, we substitute
$x_{\varphi,\lambda_{\varphi}(\alpha)}$ to be $1$ when

  1. (1)

all the polynomials occurring in $\Delta(\alpha)$ are divisors of $Q_{d}$, for
some $d|M$,

  2. (2)

all the $\lambda_{\varphi}(\alpha)$ occurring in $\Delta(\alpha)$ has highest
part to be lesser than or equal to $p^{r}$.

and put all other $x_{\varphi,\lambda_{\varphi}(\alpha)}$ to be zero. The
occurrence of the degrees follows easily. This proves the equality among both
sides. ∎

The theorems for the case of finite symplectic, orthogonal and unitary groups
will be stated without detailed proof, since the arguments will be as same as
Theorem 4.3, so we won’t be repeating them. However, we will indicate the
results from which the proofs follow.

######  Lemma 4.4.

[6] For $G(q)=\mathrm{Sp}_{2n}(q)$, we have

$\displaystyle
1+\sum\limits_{n=1}^{\infty}\dfrac{z^{2n}}{|\mathrm{Sp}_{2n}(q)|}\left(\sum\limits_{\alpha\in\mathrm{Sp}_{2n}(q)}\prod\limits_{\varphi=u\pm
1}x_{\varphi,\lambda^{\pm}_{\varphi}(\alpha)}\prod\limits_{\varphi\neq u\pm
1}x_{\varphi,\lambda_{\varphi}(\alpha)}\right)$ $\displaystyle=$
$\displaystyle\prod\limits_{\varphi=u\pm 1}\left(1+\sum\limits_{n\geq
1}\sum\limits_{\lambda^{\pm}\vdash
n}x_{\varphi,\lambda^{\pm}}\dfrac{z^{n}}{c_{\mathrm{Sp},u\pm
1,q}(\lambda^{\pm})}\right)\prod\limits_{\begin{subarray}{c}\varphi=\varphi^{*}\\\
\varphi\neq u\pm 1\end{subarray}}\left(1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}x_{\varphi,\lambda}\dfrac{(-z^{\deg\varphi})^{n}}{c_{\mathrm{GL},u-1,-(q^{\deg\varphi})^{1/2}}(\lambda)}\right)$
$\displaystyle\times$
$\displaystyle\prod\limits_{\begin{subarray}{c}\\{\varphi,\varphi^{*}\\}\\\
\varphi\neq\varphi^{*}\end{subarray}}\left(1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}x_{\varphi,\lambda}x_{\varphi^{*},\lambda}\dfrac{z^{2n\deg\varphi}}{c_{\mathrm{GL},u-1,q^{\deg\varphi}}(\lambda)}\right)$

######  Theorem 4.5.

Let $a_{n}$ denote the number of elements in $\mathrm{Sp}_{2n}(q)$ which are
$M$-th root of unity. Let $M=t\cdot p^{r}$, where $p\nmid t$. Then the
generating function of the probability $a_{n}/|\mathrm{Sp}_{2n}(q)|$ is given
by

$\displaystyle
1+\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{|\mathrm{Sp}_{2n}(q)|}z^{n}$
$\displaystyle=$ $\displaystyle\left(1+\sum\limits_{n\geq
1}\sum\limits_{\begin{subarray}{c}\lambda^{\pm}\vdash n\\\
\lambda^{\pm}_{1}\leq p^{r}\end{subarray}}\dfrac{z^{n}}{c_{\mathrm{Sp},u\pm
1,q}(\lambda^{\pm})}\right)^{o(M)}\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in D_{m}\end{subarray}}\left[1+\sum\limits_{n\geq
1}\sum\limits_{\begin{subarray}{c}\lambda\end{subarray}}\dfrac{(-z)^{n}e(d)}{c_{\mathrm{GL},u-1,-(q^{e(d)})^{1/2}}(\lambda)}\right]^{\frac{\phi(d)}{e(d)}}\right.$
$\displaystyle\left.\prod\limits_{\begin{subarray}{c}d|M\\\ d\not\in
D_{m},d\neq 1,2\end{subarray}}\left[1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}\dfrac{z^{2ne(d)}}{c_{\mathrm{GL},u-1,{q^{e(d)}}}(\lambda)}\right]^{\frac{\phi(d)}{2e(d)}}\right)$

###### Proof.

This follows from the proof of Corollary 3.8 and Proposition 3.13. The
occurrence of $o(M)$ can be justified as was done in Corollary 3.8. Finally,
we need to use Lemma 4.4. ∎

######  Lemma 4.6.

[6] Define the cycle index for sum of the both type of orthogonal groups to be

$\displaystyle 1+\sum_{n=1}^{\infty}$
$\displaystyle\left(\dfrac{z^{n}}{|\mathrm{O}^{+}_{n}(q)|}\sum\limits_{\alpha\in\mathrm{O}^{+}(q)}\prod\limits_{\varphi=u\pm
1}x_{\varphi,\lambda_{\varphi}^{\pm}(\alpha)}\prod\limits_{\varphi\neq u,u\pm
1}x_{\varphi,\lambda_{\varphi}(\alpha)}\right.$ $\displaystyle+$
$\displaystyle\left.\dfrac{z^{n}}{|\mathrm{O}^{-}_{n}(q)|}\sum\limits_{\alpha\in\mathrm{O}^{-}(q)}\prod\limits_{\varphi=u\pm
1}x_{\varphi,\lambda_{\varphi}^{\pm}(\alpha)}\prod\limits_{\varphi\neq u,u\pm
1}x_{\varphi,\lambda_{\varphi}(\alpha)}\right).$

This quantity factorises as

$\displaystyle\prod\limits_{\varphi=u\pm 1}\left(1+\sum\limits_{n\geq
1}\sum\limits_{\lambda^{\pm}\vdash
n}x_{\varphi,\lambda^{\pm}}\dfrac{z^{n}}{c_{\mathrm{O},u\pm
1,q}(\lambda^{\pm})}\right)\prod\limits_{\begin{subarray}{c}\varphi=\varphi^{*}\\\
\varphi\neq u\pm 1\end{subarray}}\left(1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}x_{\varphi,\lambda}\dfrac{(-z^{\deg\varphi})^{n}}{c_{\mathrm{GL},u-1,-(q^{\deg\varphi})^{1/2}}(\lambda)}\right)$
$\displaystyle\times$
$\displaystyle\prod\limits_{\begin{subarray}{c}\\{\varphi,\varphi^{*}\\}\\\
\varphi\neq\varphi^{*}\end{subarray}}\left(1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}x_{\varphi,\lambda}x_{\varphi^{*},\lambda}\dfrac{z^{2n\deg\varphi}}{c_{\mathrm{GL},u-1,q^{\deg\varphi}}(\lambda)}\right)$

######  Theorem 4.7.

Let $a^{\epsilon}_{n}$ denote the number of elements in
$\mathrm{O}^{\epsilon}_{n}(q)$ which are $M$-th root of unity, where
$\epsilon\in\\{\pm\\}$. Let $M=t\cdot p^{r}$, where $p\nmid t$. Then the
generating function of the sum of the probabilities
$a^{\epsilon}_{n}/|\mathrm{O}^{\epsilon}_{n}(q)|$ is given by

$\displaystyle
1+\sum\limits_{n=1}^{\infty}\left(\dfrac{a_{n}^{+}}{|\mathrm{O}^{+}_{n}(q)|}+\dfrac{a_{n}^{-}}{|\mathrm{O}^{-}_{n}(q)|}\right)z^{n}$
$\displaystyle=$ $\displaystyle\left(1+\sum\limits_{n\geq
1}\sum\limits_{\begin{subarray}{c}\lambda^{\pm}\vdash n\\\
\lambda^{\pm}_{1}\leq p^{r}\end{subarray}}\dfrac{z^{n}}{c_{\mathrm{O},u\pm
1,q}(\lambda^{\pm})}\right)^{o(M)}\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in D_{m}\end{subarray}}\left[1+\sum\limits_{n\geq
1}\sum\limits_{\begin{subarray}{c}\lambda\end{subarray}}\dfrac{(-z)^{n}e(d)}{c_{\mathrm{GL},u-1,-(q^{e(d)})^{1/2}}(\lambda)}\right]^{\frac{\phi(d)}{e(d)}}\right.$
$\displaystyle\left.\prod\limits_{\begin{subarray}{c}d|M\\\ d\not\in
D_{m},d\neq 1,2\end{subarray}}\left[1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}\dfrac{z^{2ne(d)}}{c_{\mathrm{GL},u-1,{q^{e(d)}}}(\lambda)}\right]^{\frac{\phi(d)}{2e(d)}}\right)$

###### Proof.

The reason for clubbing these two probabilities is related to how the cycle
indices for the different types of orthogonal groups are treated. This follows
from the proof of Corollary 3.9 and Proposition 3.13. The occurrence of $o(M)$
can be justified as was done in Corollary 3.9. Finally, we need to use Lemma
4.6. ∎

A formula for the generating function for the difference of the probabilities
$\dfrac{a^{+}_{n}}{|\mathrm{O}^{+}_{n}(q)|}-\dfrac{a^{-}_{n}}{|\mathrm{O}^{-}(q)|}$
can be formulated easily, and using the same techniques we can obtain a
formula for the generating function. This is ommitted here. But, a treatment
for a special case can be found in [9].

######  Lemma 4.8.

[6, 7, Theorem 10, pp 63.] For $G(q)=\mathrm{U}_{n}(q^{2})$, we have

$\displaystyle
1+\sum\limits_{n=1}^{\infty}\dfrac{z^{n}}{|\mathrm{U}_{n}(q)|}\left(\sum\limits_{\alpha\in\mathrm{U}_{n}(q^{2})}\prod\limits_{\begin{subarray}{c}\varphi\neq
u\end{subarray}}x_{\varphi,\lambda_{\varphi}(\alpha)}\right)$ $\displaystyle=$
$\displaystyle\prod\limits_{\begin{subarray}{c}\varphi\neq u\\\
\varphi=\widetilde{\varphi}\end{subarray}}\left[1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}x_{\varphi,\lambda}\dfrac{(-z)^{n\cdot\deg\varphi}}{(-q)^{\deg\varphi\cdot(\sum_{i}(\lambda_{i}^{\prime})^{2})}\prod\limits_{i\geq
1}\left(\dfrac{1}{(-q)^{\deg\varphi}}\right)_{m_{i}(\lambda_{\varphi})}}\right]$
$\displaystyle\times$
$\displaystyle\prod\limits_{\begin{subarray}{c}\varphi\neq\widetilde{\varphi}\\\
\\{\varphi,\widetilde{\varphi}\\}\end{subarray}}\left[1+\sum\limits_{n\geq
1}\sum\limits_{\lambda\vdash
n}x_{\varphi,\lambda}x_{\widetilde{\varphi},\lambda}\dfrac{z^{2n\cdot\deg\varphi}}{q^{2\deg\varphi\cdot(\sum_{i}(\lambda_{i}^{\prime})^{2})}\prod\limits_{i\geq
1}\left(\dfrac{1}{q^{2\deg\varphi}}\right)_{m_{i}(\lambda_{\varphi})}}\right]$

######  Theorem 4.9.

Let $a_{n}$ denote the number of elements in $\mathrm{U}_{n}(q^{2})$ which are
$M$-th root of unity. Let $M=t\cdot p^{r}$, where $p\nmid t$. Then the
generating function for the probability $a_{n}/|\mathrm{U}_{n}(q^{2})|$ is
given by

$\displaystyle
1+\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{|\mathrm{U}_{n}(q^{2})|}z^{n}=$
$\displaystyle\prod\limits_{m=1}^{\infty}\left(\prod\limits_{\begin{subarray}{c}d|M\\\
d\in\widetilde{D}_{m}\end{subarray}}\left[1+\sum\limits_{n\geq
1}\sum\limits_{\begin{subarray}{c}\lambda\vdash n\\\ \lambda_{1}\leq
p^{r}\end{subarray}}\dfrac{(-z)^{n\cdot
e(d)}}{(-q)^{e(d)\cdot(\sum_{i}(\lambda_{i}^{\prime})^{2})}\prod\limits_{i\geq
1}\left(\dfrac{1}{(-q)^{e(d)}}\right)_{m_{i}(\lambda_{\varphi})}}\right]^{\frac{\phi(d)}{e(d)}}\right.$
$\displaystyle\times$
$\displaystyle\left.\prod\limits_{\begin{subarray}{c}d|M\\\
d\not\in\widetilde{D}_{m}\end{subarray}}\left[1+\sum\limits_{n\geq
1}\sum\limits_{\begin{subarray}{c}\lambda\vdash n\\\ \lambda_{1}\leq
p^{r}\end{subarray}}\dfrac{z^{2n\cdot
e(d)}}{q^{2e(d)\cdot(\sum_{i}(\lambda_{i}^{\prime})^{2})}\prod\limits_{i\geq
1}\left(\dfrac{1}{q^{2e(d)}}\right)_{m_{i}(\lambda_{\varphi})}}\right]^{\frac{\phi(d)}{2e(d)}}\right),$

where $e(d)$ denotes the multiplicative order of $q$ in
$\mathbb{Z}/d\mathbb{Z}^{\times}$.

###### Proof.

This follows from the proof of Corollary 3.12 and Proposition 3.13. Finally,
we need to use Lemma 4.8. ∎

##  5\. An example

In this section, we compute the exact probility when $q\equiv-1\pmod{M}$. All
the groups are defined over $\mathbb{F}_{q}$. We concentrate mainly on the
case when $M\not=2$ is a prime and $q\equiv-1\pmod{M}$, for example,
$(q,M)=(41,7)$. Then we get

$\displaystyle x^{M}-1=(x-1)\cdot Q_{M}(x).$

Using Lemma 3.2, it will factor into $(M-1)/e(M)$ many distinct monic
polynomials of degree $e(M)=2$. In this case, the elements contributing to
$M$-th root of identity will all be semisimple (see the discussion at the
beginning of Section 3).

###  5.1. The case of $\mathrm{GL}_{n}(q)$

Let $b_{n}$ denote the proportion of $M$-th roots of identity in
$\mathrm{GL}_{n}(q)$. Then using Corollary 3.4, we get that

$\displaystyle 1+\sum\limits_{n=1}^{\infty}b_{n}z^{n}$
$\displaystyle=\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{m}}{|\mathrm{GL}_{m}(q)|}\right)\left(1+\sum\limits_{m=1}^{\infty}\dfrac{z^{2m}}{|\mathrm{GL}_{m}(q^{2})|}\right)^{{(M-1)}/{2}}.$

We now divide the computation into two cases. The first is $n$ being odd. In
this case, we should have an odd power of $z$, coming from the first term of
the product. Other contributing powers of $z$ will have all even power. Hence
probability of being an $M$-th root is

$\displaystyle\sum\limits_{\begin{subarray}{c}1\leq j\leq M\\\
j=\text{odd}\end{subarray}}\dfrac{1}{|\mathrm{GL}_{j}(q)|}\cdot\left(\sum\limits_{\lambda\vdash\frac{M-j}{2}}\prod\limits_{\ell}\dfrac{1}{|\mathrm{GL}_{\lambda_{\ell}}(q^{2})|}\right),$

where $\ell$ runs over the subscripts of the parts of
$\lambda=(\lambda_{1},\lambda_{2},\ldots)$. When $n$ is even, using the same
argument as before, we get the resulting probability to be

$\displaystyle\sum\limits_{\begin{subarray}{c}0\leq j\leq M\\\
j=\text{even}\end{subarray}}\dfrac{1}{|\mathrm{GL}_{j}(q)|}\cdot\left(\sum\limits_{\lambda\vdash\frac{M-j}{2}}\prod\limits_{\ell}\dfrac{1}{|\mathrm{GL}_{\lambda_{\ell}}(q^{2})|}\right),$

where $\ell$ runs over the subscripts of the parts of
$\lambda=(\lambda_{1},\lambda_{2},\ldots)$ and $|\mathrm{GL}_{0}(q)|$ is $1$
by convention.

###  5.2. The case of $\mathrm{Sp}_{2n}(q)$

Since $M$ is odd, we have that $o(M)=1$. We can have two possibilities, either
$M\in D_{m}$ for some $m$, or $M\not\in D_{m}$ for all $m\geq 1$. If $M\in
D_{m}$ for some $m$, then we have that the probability of an element of
$\mathrm{Sp}_{2n}(q)$ to be $M$-th root, using Corollary 3.8, is

$\displaystyle\sum\limits_{j=1}^{n}\dfrac{1}{|\mathrm{Sp}_{2j}(q)|}\cdot\left(\sum\limits_{\lambda\in\mathbf{P}_{\frac{M-1}{2}}({M-j})}\prod\limits_{\ell}\dfrac{1}{|\mathrm{U}_{\lambda_{\ell}}(q^{4})|}\right),$

and when $M\not\in D_{m}$, then the resulting probability will be

$\displaystyle\sum\limits_{j=1}^{n}\dfrac{1}{|\mathrm{Sp}_{2j}(q)|}\cdot\left(\sum\limits_{\lambda\in\mathbf{P}_{\frac{M-1}{4}}({M-j})}\prod\limits_{\ell}\dfrac{1}{|\mathrm{GL}_{\lambda_{\ell}}(q^{2})|}\right).$

The cases for the orthogonal groups $\mathrm{O}^{\epsilon}_{n}(q)$ and the
unitary group $\mathrm{U}_{n}(q^{2})$ are similar and we omit them from the
display.

## References

  * [1] Nir Avni, Tsachik Gelander, Martin Kassabov and Aner Shalev  “Word values in $p$-adic and adelic groups”  In _Bull. Lond. Math. Soc._ 45.6, 2013, pp. 1323–1330
  * [2] László Babai, Simon Guest, Cheryl E. Praeger and Robert A. Wilson  “Proportions of $r$-regular elements in finite classical groups”  In _J. Lond. Math. Soc. (2)_ 88.1, 2013, pp. 202–226
  * [3] Timothy C. Burness and Michael Giudici  “Classical groups, derangements and primes” 25, Australian Mathematical Society Lecture Series  Cambridge University Press, Cambridge, 2016, pp. xviii+346
  * [4] S. Chowla, I.. Herstein and W.. Scott  “The solutions of $x^{d}=1$ in symmetric groups”  In _Norske Vid. Selsk. Forh., Trondheim_ 25, 1952, pp. 29–31
  * [5] Veikko Ennola  “On the conjugacy classes of the finite unitary groups”  In _Ann. Acad. Sci. Fenn. Ser. A I_ 313, 1962, pp. 13
  * [6] Jason Fulman  “Cycle indices for the finite classical groups”  In _J. Group Theory_ 2.3, 1999, pp. 251–289
  * [7] Jason Fulman  “Random matrix theory over finite fields”  In _Bull. Amer. Math. Soc. (N.S.)_ 39.1, 2002, pp. 51–85
  * [8] Jason Fulman, Peter M. Neumann and Cheryl E. Praeger  “A generating function approach to the enumeration of matrices in classical groups over finite fields”  In _Mem. Amer. Math. Soc._ 176.830, 2005, pp. vi+90
  * [9] Jason Fulman, Peter M. Neumann and Cheryl E. Praeger  “A generating function approach to the enumeration of matrices in classical groups over finite fields”  In _Mem. Amer. Math. Soc._ 176.830, 2005, pp. vi+90
  * [10] Samuel Gonshaw, Martin W. Liebeck and E.. O’Brien  “Unipotent class representatives for finite classical groups”  In _J. Group Theory_ 20.3, 2017, pp. 505–525
  * [11] J.. Green  “The characters of the finite general linear groups”  In _Trans. Amer. Math. Soc._ 80, 1955, pp. 402–447
  * [12] Larry C. Grove  “Classical groups and geometric algebra” 39, Graduate Studies in Mathematics  American Mathematical Society, Providence, RI, 2002, pp. x+169
  * [13] Chun Yin Hui, Michael Larsen and Aner Shalev  “The Waring problem for Lie groups and Chevalley groups”  In _Israel J. Math._ 210.1, 2015, pp. 81–100
  * [14] Hideki Ishihara, Hiroyuki Ochiai, Yugen Takegahara and Tomoyuki Yoshida  “$p$-divisibility of the number of solutions of $x^{p}=1$ in a symmetric group”  In _Ann. Comb._ 5.2, 2001, pp. 197–210
  * [15] Tatsuhiko Koda, Masaki Sato and Yugen Takegahara  “2-adic properties for the numbers of involutions in the alternating groups”  In _J. Algebra Appl._ 14.4, 2015, pp. 155005221
  * [16] R. Kundu and A. Singh  “Generating functions for the power in $\text{GL}(n,q)$”  In _Israel Journal of Mathematics_ , 2022, pp. To appear
  * [17] Michael Larsen and Dong Quan Ngoc Nguyen  “Waring’s problem for unipotent algebraic groups”  In _Ann. Inst. Fourier (Grenoble)_ 69.4, 2019, pp. 1857–1877
  * [18] Michael Larsen and Aner Shalev  “Word maps and Waring type problems”  In _J. Amer. Math. Soc._ 22.2, 2009, pp. 437–466
  * [19] Michael Larsen and Aner Shalev  “Words, Hausdorff dimension and randomly free groups”  In _Math. Ann._ 371.3-4, 2018, pp. 1409–1427
  * [20] Michael Larsen, Aner Shalev and Pham Huu Tiep  “The Waring problem for finite simple groups”  In _Ann. of Math. (2)_ 174.3, 2011, pp. 1885–1950
  * [21] Rudolf Lidl and Harald Niederreiter  “Finite fields” With a foreword by P. M. Cohn 20, Encyclopedia of Mathematics and its Applications  Cambridge University Press, Cambridge, 1997, pp. xiv+755
  * [22] San Ling and Chaoping Xing  “Coding theory” A first course  Cambridge University Press, Cambridge, 2004, pp. xii+222
  * [23] I.. Macdonald  “Numbers of conjugacy classes in some finite classical groups”  In _Bull. Austral. Math. Soc._ 23.1, 1981, pp. 23–48
  * [24] Gunter Malle and Donna Testerman  “Linear algebraic groups and finite groups of Lie type” 133, Cambridge Studies in Advanced Mathematics  Cambridge University Press, Cambridge, 2011, pp. xiv+309
  * [25] Alice C. Niemeyer, Tomasz Popiel and Cheryl E. Praeger  “Abundant $p$-singular elements in finite classical groups”  In _J. Algebra_ 408, 2014, pp. 189–204
  * [26] Alice C. Niemeyer, Tomasz Popiel and Cheryl E. Praeger  “Algorithms to identify abundant $p$-singular elements in finite classical groups”  In _Bull. Aust. Math. Soc._ 86.1, 2012, pp. 50–63
  * [27] Alice C. Niemeyer and Cheryl E. Praeger  “Estimating proportions of elements in finite groups of Lie type”  In _J. Algebra_ 324.1, 2010, pp. 122–145
  * [28] Alice C. Niemeyer and Cheryl E. Praeger  “On permutations of order dividing a given integer”  In _J. Algebraic Combin._ 26.1, 2007, pp. 125–142
  * [29] Saikat Panja and Anupam Singh  “Powers in finite orthogonal and symplectic groups: A generating function approach”  In _Israel Journal of Mathematics_ , 2023, pp. To appear
  * [30] Saikat Panja and Anupam Singh  “Powers in finite unitary groups”, 2023  arXiv:2304.13735 [math.GR]
  * [31] Aner Shalev  “Word maps, conjugacy classes, and a noncommutative Waring-type theorem”  In _Ann. of Math. (2)_ 170.3, 2009, pp. 1383–1416
  * [32] Ken-ichi Shinoda  “The characters of Weil representations associated to finite fields”  In _J. Algebra_ 66.1, 1980, pp. 251–280
  * [33] D.. Taylor  “Conjugacy Classes in Finite Orthogonal Groups”  online notes, 2020  URL: https://www.maths.usyd.edu.au/u/don/code/Magma/GOConjugacy
  * [34] D.. Taylor  “Conjugacy Classes in Finite Symplectic Groups”  online notes, 2020  URL: https://www.maths.usyd.edu.au/u/don/code/Magma/SpConjugacy
  * [35] D.. Taylor  “Conjugacy Classes in Finite Unitary Groups”  online notes, 2020  URL: https://www.maths.usyd.edu.au/u/don/code/Magma/GUConjugacy.pdf
  * [36] G.. Wall  “On the conjugacy classes in the unitary, symplectic and orthogonal groups”  In _J. Austral. Math. Soc._ 3, 1963, pp. 1–62
  * [37] Joseph L. Yucas and Gary L. Mullen  “Self-reciprocal irreducible polynomials over finite fields”  In _Des. Codes Cryptogr._ 33.3, 2004, pp. 275–281