Euler characteristics and p-singular elements in finite groups

TL;DR

This paper explores the relationship between Euler characteristics of orbit categories in finite groups and classical theorems in group theory, establishing new equivalences among them.

Contribution

It introduces a novel approach linking Euler characteristics to fundamental theorems of Frobenius, Brown, Steinberg, and Solomon in finite group theory.

Findings

Established equivalences between Frobenius and Brown's theorems

Linked Steinberg and Solomon's theorems via Euler characteristics

Provided a new perspective on p-singular elements in finite groups

Abstract

We use the Euler characteristic of the orbit category of a finite group to establish equivalences between theorems of Frobenius and K.S. Brown and between theorems of Steinberg and L. Solomon.

Equations

$$G_p=\{g\in G\mid g^{|G{|}_p}=e\}=\bigcup {\mathrm{Syl}}_p(G)$$

$$n\mid |G|⟹n\mid |\{g\in G\mid g^n=e\}|$$

$$\sum _{n=1}^{\infty }|({\Sigma }_n{)}_p|\frac{x^n}{n!}=\exp (x+\frac{x^p}{p}+\frac{x^{p^2}}{p^2}+\cdots +\frac{x^{p^m}}{p^m}+\cdots )$$

$$|G_p|=p^{-1}+\sum _{1\le C\le G}(1-p^{-1})|C|$$

$$[{\mathcal{O}}_G^{p+\mathrm{rad}}]([H],[K])=|{\mathcal{O}}_G(H,K)|,[H],[K]\in [{\mathcal{S}}_G^{p+\mathrm{rad}}]$$

$$|G_p|=|G|\chi ({\mathcal{O}}_G^{p+\mathrm{rad}})=\sum _{K\in {\mathcal{S}}_G^{p+\mathrm{rad}}}-\tilde{\chi }({\mathcal{S}}_{{\mathcal{O}}_G(K)}^{p+\ast })|K|$$

$$\sum _{[K]\in [{\mathcal{S}}_G^{p+\mathrm{rad}}]}|{\mathcal{O}}_G(H,K)|k_{[\mathcal{O}]}^K=1$$

$$k_{\mathcal{O}}^K=\frac{|N_G(K)|}{|G|}{k}_{[\mathcal{O}]}^{[K]},k_{\mathcal{S}}^K=\frac{|G|}{|K|}{k}_{\mathcal{O}}^K=\frac{|G|}{|K|}\frac{|N_G(K)|}{|G|}{k}_{[\mathcal{O}]}^K=|{\mathcal{O}}_G(K)|k_{[\mathcal{O}]}^K$$

$$\sum _{[K]\in [{\mathcal{S}}_G^{p+\mathrm{rad}}]}\frac{|{\mathcal{O}}_G(H,K)|}{|{\mathcal{O}}_G(K)|}{k}_{\mathcal{S}}^K=1$$

$$[[{\mathcal{O}}_G^{p+\mathrm{rad}}]]([H],[K])=\frac{|{\mathcal{O}}_G(H,K)|}{|{\mathcal{O}}_G(K)|},[H],[K]\in [{\mathcal{S}}_G^{p+\mathrm{rad}}]$$

$$\frac{|{\mathcal{O}}_G(H,K)|}{|{\mathcal{O}}_G(K)|}=|N_G(H,K)/N_G(K)|=|\{L\in [K]\mid H\supseteq L\}|$$

$$[{\mathcal{O}}_{{\Sigma }_4}^{2+\mathrm{rad}}]=\left(\begin{array}{cc}1& 0\\ 3& 6\end{array}\right),[[{\mathcal{O}}_{{\Sigma }_4}^{2+\mathrm{rad}}]]=\left(\begin{array}{cc}1& 0\\ 3& 1\end{array}\right)$$

$$|G_p|+\tilde{\chi }({\mathcal{S}}_G^{p+\ast })+\sum _{[H]\ne 1}\frac{\tilde{\chi }({\mathcal{S}}_{{\mathcal{O}}_G(H)}^{p+\ast })}{|{\mathcal{O}}_G(H){|}_p}\frac{|G|}{|{\mathcal{O}}_G(H){|}_{p^{\prime }}}=0$$

$$[[{\mathcal{O}}_K^{p+\mathrm{rad}}]](U_I,U_J)=\{\begin{array}{ll}|P_I\text{mathchar}58P_J|& I\supseteq J\\ 0& \text{otherwise}\end{array}$$

$$\left(\begin{array}{cccc}1& 0& 0& 0\\ 3& 1& 0& 0\\ 3& 0& 1& 0\\ 21& 7& 7& 1\end{array}\right)\left(\begin{array}{c}1\\ -2\\ -2\\ 8\end{array}\right)=\left(\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right)$$

$$\begin{gathered} |(P_I{)}_p|=\sum _{I\supseteq J}-\tilde{\chi }({\mathcal{S}}_{L_J}^{p+\ast })|P_I\text{mathchar}58P_J||U_J|=\sum _{I\supseteq J}(-1{)}^{|J|}|P_I\text{mathchar}58P_J||L_J{|}_p|U_J|=|K{|}_p\sum _{I\supseteq J}(-1{)}^{|J|}|P_I\text{mathchar}58P_J| \\ \stackrel{\text{Corleavevmodenobreak refcor:solomon}}{=}|K{|}_p|L_I{|}_p=|K{|}_p|P_I\text{mathchar}58U_I{|}_p=|K{|}_p^2/|U_I| \end{gathered}$$

$$\sum _{J\subseteq \Pi }(-1{)}^J\frac{W_{\Pi }(q)}{W_J(q)}=q^{|{\Sigma }^{+}|},\sum _{J\subseteq \Pi }(-1{)}^J\frac{W_{\Pi }(q)}{W_J(q)}{q}^{|{\Sigma }_J^{+}|}=1$$

$$\sum (-1{)}^k(\genfrac{}{}{0ex}{}{[m](q)}{[m_1](q),\cdots ,[m_k](q)})=(-1{)}^m{q}^{\left(\genfrac{}{}{0ex}{}{m}{2}\right)},\sum (-1{)}^k(\genfrac{}{}{0ex}{}{[m](q)}{[m_1](q),\cdots ,[m_k](q)})q^{\sum \left(\genfrac{}{}{0ex}{}{m_i}{2}\right)}=(-1{)}^m$$

$$\sum \frac{(-1{)}^k\prod _{d=m_k}^{m-1}[2d](q)}{[m_1]!(q)\cdots [m_{k-1}]!(q)}=(-1{)}^m{q}^{(m-1{)}^2},\sum \frac{(-1{)}^k\prod _{d=m_k}^{m-1}[2d](q)}{[m_1]!(q)\cdots [m_{k-1}]!(q)}{q}^{{\sum }_{i=1}^{m_{k-1}}\left(\genfrac{}{}{0ex}{}{m_i}{2}\right)+(m_k-1{)}^2}=(-1{)}^m$$

$$\sum \frac{(-1{)}^k\prod _{d=1}^{2m}[d]((-1{)}^{d}q)}{[m_1]!(q^2)\cdots [m_k]!(q^2)}-\sum \frac{(-1{)}^k\prod _{d=2m_k+2}^{2m}[d]((-1{)}^{d}q)}{[m_1]!(q^2)\cdots [m_{k-1}]!(q^2)}=(-1{)}^m{q}^{\left(\genfrac{}{}{0ex}{}{2m}{2}\right)}$$

$$\sum \frac{(-1{)}^k\prod _{d=1}^{2m}[d]((-1{)}^{d}q)}{[m_1]!(q^2)\cdots [m_k]!(q^2)}{q}^{\sum \left(\genfrac{}{}{0ex}{}{m_i}{2}\right)}-\sum \frac{(-1{)}^k\prod _{d=2m_k+2}^{2m}[d]((-1{)}^{d}q)}{[m_1]!(q^2)\cdots [m_{k-1}]!(q^2)}{q}^{\sum \left(\genfrac{}{}{0ex}{}{m_i}{2}\right)}=(-1{)}^m$$

$$\sum \frac{(-1{)}^k\prod _{d=1}^{2m+1}[d]((-1{)}^{d}q)}{[m_1]!(q^2)\cdots [m_k]!(q^2)}-\sum \frac{(-1{)}^k\prod _{d=2m_k+2}^{2m+1}[d]((-1{)}^{d}q)}{[m_1]!(q^2)\cdots [m_{k-1}]!(q^2)}=(-1{)}^m{q}^{\left(\genfrac{}{}{0ex}{}{2m+1}{2}\right)}$$

$$\sum \frac{(-1{)}^k\prod _{d=1}^{2m+1}[d]((-1{)}^{d}q)}{[m_1]!(q^2)\cdots [m_k]!(q^2)}{q}^{\sum \left(\genfrac{}{}{0ex}{}{m_i}{2}\right)}-\sum \frac{(-1{)}^k\prod _{d=2m_k+2}^{2m+1}[d]((-1{)}^{d}q)}{[m_1]!(q^2)\cdots [m_{k-1}]!(q^2)}{q}^{\sum \left(\genfrac{}{}{0ex}{}{m_i}{2}\right)}=(-1{)}^m$$

$$\varnothing$$

$$S(A_{2m-1})$$

$$S(A_{2m})$$

$$P(A_m)(q)=\prod _{1\le d\le m+1}[d]((-1{)}^{d}q),P(a_{2m-1})=\prod _{1\le d\le m}[d](q^2)=[m]!(q^2)m\ge 1$$

$$P(S(A_m))=\{(-1{)}^{|\Pi (a)/C_2|}P(a)(q)\mid a\in S(A_m)\}$$

$$|{\mathrm{GL}}_n({\mathbf{F}}_q{)}_p/{\mathrm{GL}}_n({\mathbf{F}}_q)|=\frac{1}{n!}\sum _{\lambda ⊢n}T(\lambda )\prod _{b\in \lambda }(q^b-1{)}_p$$

Full text

Supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92)

1. Introduction

Let $G$ be a finite group with unit element $e$, $p$ a prime number and $|G|_{p}$ the $p$-part of the group order. An element of $G$ is $p$-singular if its order is a power of $p$ [6, Definition 40.2, §82.1]. Write

[TABLE]

for the set of all $p$-singular elements in $G$. In other words, $G_{p}$ is the solution set in $G$ to the equation $X^{|G|_{p}}=1$ or the union of all the Sylow $p$-subgroups of $G$. A theorem of Frobenius from $1907$, or even earlier,

[TABLE]

contains as a special case a basic fact about the number, $|G_{p}|$, of $p$-singular group elements [7, 12] [6, Corollary 41.11] [18, 11.2, Corollary 2].

Theorem 1.1** (Frobenius $1907$).**

$|G|_{p}\mid|G_{p}|$**

The number of $p$-singular elements is known for the symmetric groups and for the finite groups of Lie type in defining characteristic $p$:

• •

The exponential generating function for the number of $p$-singular permutations in the symmetric groups $\Sigma_{n}$ is [21, Example 5.2.10]

[TABLE]

• •

$|K_{p}|=|K|_{p}^{2}$ for a finite group $K$ of Lie type in defining characteristic $p$ (Theorem 4.1)

The aim of this note is to relate Frobenius’ theorem (Theorem 1.1) to a theorem of K.S. Brown (Theorem 3.1) and Steinberg’s theorem (Theorem 4.1) to a theorem of L. Solomon (Theorem 4.2). These two pairs of theorems are linked by the Euler characteristic of the orbit category discussed in Proposition 2.3.(1).

The following notation will be used in this note:

[TABLE]

If $\mathcal{C}_{G}$ is a category whose objects are all subgroups of $G$, then $\mathcal{C}_{G}^{p}$, $\mathcal{C}_{G}^{p+*}$, $\mathcal{C}_{G}^{p+\mathrm{rad}}$ denotes the full subcategory of $\mathcal{C}_{G}$ on all $p$-subgroups, non-trivial $p$-subgroups, $p$-radical $p$-subgroups, respectively. (A $p$-subgroup $H$ of $G$ is $p$-radical if $H=O_{p}N_{G}(H)$.)

2. Using Euler characteristics to count $p$-singular elements

We apply Tom Leinster’s theory of Euler characteristics of finite categories [14] to the orbit category $\mathcal{O}_{G}^{p}$.

Let $\zeta$ be a square matrix with rational coeffecients. A weighting for $\zeta$ is a vector $k$ such that all coordinates of $\zeta k$ equal $1$. A coweighting is a weighting for the transpose of $\zeta$. The matrix $\zeta$ has Euler characteristic if it admits both a weighting and a coweighting, and the Euler characteristic of $\zeta$, $\chi(\zeta)$, is then the coordinate sum of a weighting or a coweighting [14, Lemma 2.1, Definition 2.2]. The Euler characteristic of an invertible matrix is the sum of the entries of the inverse.

Let $\mathcal{C}$ be a finite category. The $\zeta$-matrix of $\mathcal{C}$ is the square matrix $\zeta(\mathcal{C})=\left(|\mathcal{C}(a,b)|\right)_{a,b\in\mathrm{Ob}(\mathcal{C})}$ recording the cardinalities of all the morphism sets in $\mathcal{C}$. A weighting or coweighting for $\mathcal{C}$ is a weighting or coweighting for $\zeta(\mathcal{C})$. The Euler characteristic of $\mathcal{C}$ is $\chi(\mathcal{C})=\chi(\zeta(\mathcal{C}))$ when $\mathcal{C}$ has a weighting and a coweighting. The reduced Euler characteristic of $\mathcal{C}$ is $\widetilde{\chi}(\mathcal{C})=\chi(\mathcal{C})-1$. The Euler characteristic of any finite category with an initial or terminal object is $1$.

The finite categories of this note all admit weightings and coweightings. By the weighting for e.g. $\mathcal{O}_{G}^{p}$ we mean the weighting that is constant on isomorphism classes of objects [8, p 3035].

Lemma 2.1**.**

The number of $p$-singular elements in $G$ is

[TABLE]

where the sum is over all cyclic $p$-subgroups $C$ of $G$.

Proof.

Declare two $p$-singular elements to be equivalent of they generate the same cyclic subgroup. The set of equivalence classes is the set of cyclic $p$-subgroups $C$ of $G$. The number of elements in the equivalence class $C$ is the number of generators of $C$: $p^{-1}+(1-p^{-1})|C|$ if $|C|=1$ and $(1-p^{-1})|C|$ if $|C|>1$. ∎

The weightings for the poset $\mathcal{S}_{G}^{p}$ and the category $\mathcal{O}_{G}^{p}$, $k^{K}_{\mathcal{S}}=-\widetilde{\chi}(\mathcal{S}_{\mathcal{O}_{G}(K)}^{p+*})$ and $k^{K}_{\mathcal{O}}=-\frac{1}{|G|}\widetilde{\chi}(\mathcal{S}_{\mathcal{O}_{G}(K)}^{p+*})|K|$ [13, Theorem 1.3], vanish off the $p$-radical subgroups by Quillen’s [17, Proposition 2.4]. Thus the weightings for $\mathcal{S}_{G}^{p}$, $\mathcal{O}_{G}^{p}$ restrict to weightings for the full subcategories $\mathcal{S}_{G}^{p+\mathrm{rad}}$, $\mathcal{O}_{G}^{p+\mathrm{rad}}$ and $\chi(\mathcal{S}_{G}^{p+\mathrm{rad}})=\chi(\mathcal{S}_{G}^{p})=1$, $\chi(\mathcal{O}_{G}^{p+\mathrm{rad}})=\chi(\mathcal{O}_{G}^{p})$ [13, Lemma 2.9].

It can be more convenient to work with conjugacy classes of subgroups rather than the subgroups themselves. Let $[\mathcal{S}_{G}^{p+\mathrm{rad}}]$ be the set of conjugcacy classes $[K]$ of $p$-radical subgroups $K$ of $G$. Since $|\mathcal{O}_{G}(H,K)|=|(K\backslash G)^{H}|$ is the mark of $H$ on the transitive right $G$-set $K\backslash G$, the square matrix $[\mathcal{O}_{G}^{p+\mathrm{rad}}]$ with entries

[TABLE]

is Burnside’s table of marks [3] for the $p$-radical subgroup classes. It is easy to see that the category $\mathcal{O}_{G}^{p+\mathrm{rad}}$ and the table of marks $[\mathcal{O}_{G}^{p+\mathrm{rad}}]$ have the same Euler characteristic [13, §2.4].

Proposition 2.3**.**

Let $G$ be a finite group and $p$ a prime number.

1. (1)

$\displaystyle\sum\limits_{K\in\mathcal{S}_{G}^{p+\mathrm{rad}}}-\widetilde{\chi}(\mathcal{S}_{\mathcal{O}_{G}(K)}^{p+*})|K|=|G_{p}|$** 2. (2)

$\sum\limits_{K\in\mathcal{S}_{G}^{p+\mathrm{rad}}}-\widetilde{\chi}(\mathcal{S}_{\mathcal{O}_{G}(K)}^{p+*})=1$** 3. (3)

$\sum\limits_{[K]\in[\mathcal{S}_{G}^{p+\mathrm{rad}}]}-\widetilde{\chi}(\mathcal{S}_{\mathcal{O}_{G}(K)}^{p+*})\frac{|\mathcal{O}_{G}(H,K)|}{|\mathcal{O}_{G}(K)|}=1$* for any $p$-radical subgroup $H$*

Proof.

Lemma 2.1 combined with [13, Theorem 1.3.(4)] show that

[TABLE]

where the sum ranges over all $p$-radical subgroups $K$ of $G$. This proves (1). Item (2) simply expresses that $\mathcal{S}_{G}^{p+\mathrm{rad}}$, with $O_{p}(G)$ as its least element [8, Proposition 6.3], has Euler characteristic equal to $1$.

The weighting, $k^{\bullet}_{[\mathcal{O}]}\colon[\mathcal{S}_{G}^{p+\mathrm{rad}}]\to\mathbf{Q}$, for the table of marks of the $p$-radical subgroup classes (2.2) satisfies

[TABLE]

for all $p$-radical subgroups $H$. The weightings, $k^{\bullet}_{\mathcal{O}}$ and $k^{\bullet}_{\mathcal{S}}$, for $\mathcal{O}_{G}^{p+\mathrm{rad}}$ and $\mathcal{S}_{G}^{p+\mathrm{rad}}$ are [13, Proposition 2.14]

[TABLE]

and therefore

[TABLE]

which is the third item. ∎

Proposition 2.3.(1) expresses that $|G_{p}|=|G|\chi(\mathcal{O}_{G}^{p+\mathrm{rad}})=|G|\chi([\mathcal{O}_{G}^{p+\mathrm{rad}}])$ can be computed from the table of marks for the $p$-radical subgroups (2.2).

The content of Proposition 2.3.(3) is that the vector $(k_{\mathcal{S}}^{K})_{K\in[\mathcal{S}_{G}^{p+\mathrm{rad}}]}$ is a weighting for $[[\mathcal{O}_{G}^{p+\mathrm{rad}}]]$, the modified table of marks, defined to be the square matrix with entries

[TABLE]

In other words, $k^{K}_{\mathcal{S}}=k^{K}_{[[\mathcal{O}]]}$ for all $p$-radical subgroups $K$ where $k^{\bullet}_{[[\mathcal{O}]]}$ is the weighting for $[[\mathcal{O}_{G}^{p+\mathrm{rad}}]]$.

The normalizer $N_{G}(K)$ acts on the transporter set $N_{G}(H,K)=\{g\in G\mid H^{g}\subseteq K\}$ and the orbit set corresponds bijectively via the map $g\to K^{g^{-1}}$ to the set $\{L\in[K]\mid H\supseteq L\}$ of conjugates of $K$ containing $H$. Thus the modified mark

[TABLE]

is the number of $H$-supergroups conjugate to $K$.

Example 2.5** ($G=\Sigma_{4}$, $p=2$).**

The $2$-radical subgroup classes of the symmetric group $\Sigma_{4}$ are the Sylow $2$-subgroup $D_{8}$ and $O_{2}(\Sigma_{4})=C_{2}\times C_{2}$ of order $4$. The table of marks (2.2) and the modified table of marks (2.4) for the $2$-radical subgroup classes in $\Sigma_{4}$ are

[TABLE]

The weighting for the table of marks is $k_{[\mathcal{O}]}=(1,-1/3)$ and $|\Sigma_{4}|\chi([\mathcal{O}_{\Sigma_{4}}^{2+\mathrm{rad}}])=24(1-1/3)=16$ is the number of $2$-singular elements in $\Sigma_{4}$ in agreement with Proposition 2.3.(1). The modified table of marks has weighting $k_{[[\mathcal{O}]]}=(1,-2)$ which by Proposition 2.3.(3) means that $k^{D_{8}}_{\mathcal{S}}=1$ and $k^{C_{2}\times C_{2}}_{\mathcal{S}}=-2$. Since the subgroups $D_{8}$, $C_{2}\times C_{2}$ have lengths $3$, $1$, the Euler characteristic of the Brown poset $\mathcal{S}_{\Sigma_{4}}^{2+\mathrm{rad}}$ is $\chi(\mathcal{S}_{\Sigma_{4}}^{2+\mathrm{rad}})=3\cdot 1+1\cdot(-2)=1$ in agreement with Proposition 2.3.(2).

3. The theorems of Frobenius and Brown for finite groups are equivalent

The following theorem was proved by K.S. Brown [1] (and reproved by Quillen [17, Corollary 4.2], Webb [23, Theorem 8.1] and others).

Theorem 3.1** (Brown $1975$).**

$|G|_{p}\mid\widetilde{\chi}(\mathcal{S}_{G}^{p+*})$**

It was observed in [2, 11] that Möbius functions link the theorems of Frobenius and Brown. We here note that also Proposition 2.3.(1) connects the two theorems.

Proposition 3.2**.**

Theorems 1.1 and 3.1 are equivalent given Proposition 2.3.(1).

Proof.

Proposition 2.3.(1) may be rewritten on the form

[TABLE]

where we have isolated the contribution from the trivial subgroup and the sum is over classes of non-trivial $p$-radical subgroups of $G$.

Assume first that Theorem 1.1 holds. In Equation (3.3), we may assume that

• •

$\widetilde{\chi}(\mathcal{S}_{\mathcal{O}_{G}(H)}^{p+*})/|\mathcal{O}_{G}(H)|_{p}$ is an integer when $H$ is nontrivial (as part of an inductional argument)

• •

$|G|/|\mathcal{O}_{G}(H)|_{p^{\prime}}$ is an integer divisible by $|G|_{p}$ (as $|\mathcal{O}_{G}(H)|$ divides $|G|$)

Thus every term in the sum is divisible by $|G|_{p}$ and so is $|G_{p}|$ by assumption. We conclude that $\widetilde{\chi}(\mathcal{S}_{G}^{p+*})$ is divisible by $|G|_{p}$ and we have arrived at Theorem 3.1.

Assume next that Theorem 3.1 holds. In Equation (3.3)

• •

$|G|/|\mathcal{O}_{G}(H)|_{p^{\prime}}$ is an integer divisible by $|G|_{p}$

• •

$\widetilde{\chi}(\mathcal{S}_{\mathcal{O}_{G}(H)}^{p+*})/|\mathcal{O}_{G}(H)|_{p}$ is an integer

• •

$\widetilde{\chi}(\mathcal{S}_{G}^{p+*})$ is divisible by $|G|_{p}$

and thus $|G|_{p}$ divides for $|G_{p}|$. This is Theorem 1.1. ∎

4. The theorems of Solomon and Steinberg

for finite groups of Lie type are equivalent

Let $\Sigma$ be a reduced and crystallographic root system with fundamental and positive roots $\Pi,\Sigma^{+}\subseteq\Sigma$ [10, Definition 1.8.1]. Suppose $\overline{K}(\Sigma)$ is a semisimple $\overline{\mathbf{F}}_{p}$-algebraic group with root system $\Sigma$ [10, Theorem 1.10.4] equipped with a (standard form) Steinberg endomorphism $\sigma$ [10, Definition 1.15.(b), Remarks 2.2.5.(e)]. Assuming $\Sigma$ to be also irreducible [10, Definition 1.8.4], let $K=O^{p^{\prime}}C_{\overline{K}(\Sigma)}(\sigma)$ be the finite group in $\mathit{Lie}(p)$ with $\sigma$-setup $(\overline{K}(\Sigma),\sigma)$ [10, Definition 2.2.2].

The number of $p$-singular elements in $K$ was determined by Steinberg [22, 15.2].

Theorem 4.1** (Steinberg $1968$).**

$|K_{p}|=|K|_{p}^{2}$**

The surjections $\Sigma\to\widetilde{\Sigma}\to\widehat{\Sigma}$ of [10, (2.3.1)] induce surjections $\Pi\to\widetilde{\Pi}\to\widehat{\Pi}$ of sets. Here, $\widetilde{\Sigma}$ is the twisted root system of $K$ [10, p 41], and $\widehat{\Sigma}=\widetilde{\Sigma}/\!\!\sim$ is the set of equivalence classes of twisted roots pointing in the same direction. If $K$ is an untwisted group of Lie type [10, Definition 2.2.4], $\Sigma=\widetilde{\Sigma}=\widehat{\Sigma}$.

For every subset $J\subseteq\widehat{\Pi}$ we have associated subgroups $P_{J},U_{J},L_{J}\subseteq K$ such that $U_{J}=O_{p}P_{J}$, $P_{J}=N_{K}(U_{J})$ and $P_{J}=U_{J}\rtimes L_{J}$ [10, Theorem 2.6.5]. The $P_{J}$ are parabolic subgroups, the $U_{J}$ are unipotent $p$-radical subgroups and the $L_{J}$ are Levi complements [10, Definition 2.6.4, Definition 2.6.6]. The extreme cases where $J=\emptyset,\widehat{\Pi}$ are, $P_{\emptyset}=U_{\emptyset}\rtimes L_{\emptyset}$, where $P_{\emptyset}=B$ is a Borel subgroup of $K$, $U_{\emptyset}=U$ a Sylow $p$-subgroup [10, p 41, Theorems 2.3.4, 2.3.7] and $L_{\emptyset}=H$ is a maximal torus or Cartan subgroup [10, Theorem 2.4.7, Definition 2.4.12], and $P_{\widehat{\Pi}}=K=L_{\widehat{\Pi}}$, $U_{\widehat{\Pi}}=1$.

The following polynomial identity dates back to L. Solomon [19, Corollary 1.1] [4, Theorem 9.4.5, §14].

Theorem 4.2** (Solomon $1966$).**

$\sum\limits_{J\subseteq\widehat{\Pi}}(-1)^{|J|}|P_{\widehat{\Pi}}\mathrel{\mathop{\mathchar 58\relax}}P_{J}|=|K|_{p}$**

Solomon’s theorem, essentially a statement about reflection groups, generalizes to the following two identities that are Möbius inverses to each other.

Corollary 4.3**.**

For any subset $I$ of $\widehat{\Pi}$, $\sum\limits_{I\supseteq J}(-1)^{|J|}|P_{I}\mathrel{\mathop{\mathchar 58\relax}}P_{J}|=|L_{I}|_{p}$ and $\sum\limits_{I\supseteq J}(-1)^{|J|}|P_{I}\mathrel{\mathop{\mathchar 58\relax}}P_{J}||L_{J}|_{p}=1$.

We use a consequence of the Borel–Tits theorem [10, Theorem 3.1.3] to determine the modified table of marks for the $p$-radical subgroups of $K$.

Lemma 4.4**.**

The modified table of marks (2.4) for the $p$-radical subgroups, $U_{I}$, $I\subseteq\widehat{\Pi}$, of $K$ has entries

[TABLE]

for all subsets $I,J\subseteq\widehat{\Pi}$.

Proof.

By [10, Corollary 3.16], assisted by [10, Theorem 2.6.7] to show that any parabolic subgroup containing $U$ contains $B$, the set $\{U_{I}\mid I\subseteq\widehat{\Pi}\}$ is the Alperin–Goldschmidt conjugation family [9, Theorem 16.1] controlling fusion in $U$. It follows that if $g\in K$ conjugates $U_{I}$ into some $U_{J}$ then $U_{I}^{g}=U_{I}$ so that $I\supseteq J$ and $g\in N_{K}(U_{I})=P_{I}$ [10, Theorem 2.6.5]. This shows that the transporter set $N_{K}(U_{I},U_{J})$ equals $N_{K}(U_{I})$ in case $I$ contains $J$ and is empty otherwise. ∎

This leads to a weak version of the Solomon–Tits theorem [5, Corollary 7.3].

Corollary 4.5**.**

$-\widetilde{\chi}(\mathcal{S}_{L_{I}}^{p+*})=(-1)^{|I|}|L_{I}|_{p}$* for all $I\subseteq\widehat{\Pi}$.*

Proof.

This follows immediately from Proposition 2.3.(3) and the second identity of Corollary 4.3 as the coefficients $|P_{I}\mathrel{\mathop{\mathchar 58\relax}}P_{J}|$ are the the modified marks by Lemma 4.4. ∎

Example 4.6**.**

The group $\operatorname{GL}_{3}(\mathbf{F}_{2})\in\mathit{Lie}(2)$ has fundamental roots $\Pi=\{\alpha_{1},\alpha_{2}\}$. Its parabolic subgroups, $P_{\emptyset}$, $P_{\{\alpha_{1}\}}$, $P_{\{\alpha_{2}\}}$, $P_{\Pi}$, have orders $8,8\cdot 3,8\cdot 3,8\cdot 21$ and Levi complements, $1$, $\operatorname{GL}_{2}(\mathbf{F}_{2})$, $\operatorname{GL}_{2}(\mathbf{F}_{2})$, $\operatorname{GL}_{3}(\mathbf{F}_{2})$, with Sylow $2$-subgroup orders $1,2,2,8$. The signed vector $(1,-2,-2,8)$ of Corollary 4.5 is indeed the weighting for the modified table of marks of Lemma 4.4 as

[TABLE]

By Proposition 2.3.(1), this weighting for the modified table of marks records the negative reduced Euler characteristics, $-\widetilde{\chi}(\mathcal{S}_{L_{I}}^{2+*})$, of the Brown posets for the Levi complements.

We are now prepared to prove a version of Steinberg’s theorem valid for all parabolic subgroups of $K$.

Theorem 4.7**.**

$|P_{p}||O_{p}(P)|=|K|_{p}^{2}$* for any parabolic subgroup $P$ of $K$.*

Proof.

There is always a bjiection between the $p$-radical subgroups of a finite group $G$ and those of $G/O_{p}(G)$ [8, Proposition 6.3]. In particular, $\{U_{J}\mid I\supseteq J\}$ is a complete set of representatives for the $p$-radical subgroups of $P_{I}$ corresponding to the $p$-radical subgroup classes of $L_{I}$. Obviously, $N_{P_{I}}(U_{J})=P_{I}\cap N_{K}(U_{J})=P_{I}\cap P_{J}=P_{J}=N_{K}(U_{J})$, $\mathcal{O}_{P_{I}}(U_{J})=P_{J}/U_{J}=L_{J}=\mathcal{O}_{K}(U_{J})$ and $|P_{I}\mathrel{\mathop{\mathchar 58\relax}}N_{P_{I}}(U_{J})|=|P_{I}\mathrel{\mathop{\mathchar 58\relax}}P_{J}|$ is the length of $U_{J}$ in $P_{I}$ and $K$. By Proposition 2.3.(1), the number of $p$-singular elements in $P_{I}$ is

[TABLE]

This finishes the proof as $U_{I}=O_{p}(P_{I})$. ∎

Theorems 4.1 and 4.7 apply e.g. to $\Omega_{2m}^{\pm}(\mathbf{F}_{q})$, $P\Omega_{2m}^{\pm}(\mathbf{F}_{q})$, $\operatorname{Spin}^{\pm}_{2m}(\mathbf{F}_{q})$ for all prime powers $q$, to $\operatorname{SO}^{\pm}_{2m}(\mathbf{F}_{q})$ for odd $q$, but not to $\operatorname{SO}^{\pm}_{2m}(\mathbf{F}_{2^{e}})$ [10, §2.7]. ($\operatorname{SO}^{-}_{4}(\mathbf{F}_{2})\cong\Sigma_{5}$ of order $8\cdot 15$ contains $56<8^{2}$ $2$-singular elements.) They also apply to $\Omega_{2m+1}(\mathbf{F}_{q})$, $\operatorname{SO}_{2m+1}(\mathbf{F}_{q})$ and $\operatorname{Spin}_{2m+1}(\mathbf{F}_{q})$ for all $q$.

The $q$-bracket of the natural number $d$ is the polynomial $[d](q)=q^{d-1}+\cdots+q+1\in\mathbf{Z}[q]$ of degree $d-1$ with value $[d](1)=d$ at $q=1$. For a reflection group $W$, put $W(q)=\prod_{d}[d](q)$ where the product is over the degrees $d$ of the basic polynomial invariants [4, Proposition 10.2.5]. The two identities of Corollary 4.3 with $I=\Pi$ for the Chevalley (untwisted) groups associated to $\overline{K}(\Sigma)$ with Weyl group $W$,

[TABLE]

are two $q$-analogs of Witt’s identity [24] [19, (5)] $\sum\limits_{J\subseteq\Pi}(-1)^{J}|W_{\Pi}\mathrel{\mathop{\mathchar 58\relax}}W_{J}|=1$.

Let $\mathrm{OP}(m)=\{(m_{1},\ldots,m_{k})\mid k\geq 1,m_{i}\geq 1,\sum m_{i}=m\}$ denote the set of all the $2^{m-1}$ ordered partitions of $m$ [20, p 14].

Example 4.9**.**

Subsystems of the root systems $A_{m-1}$ or $B_{m-1}$ are indexed by $\mathrm{OP}(m)$ via the bijection taking $(m_{1},\ldots,m_{k})\in\mathrm{OP}(m)$ to $A_{m_{1}-1}\times\cdots\times A_{m_{k}-1}$ or $A_{m_{1}-1}\times\cdots\times A_{m_{k-1}-1}\times B_{m_{k}-1}$ (where $A_{0}$ is the empty root system and $B_{0}=A_{0}$, $B_{1}=A_{1}$). The incarnations of equations (4.8) for the Chevalley groups $\operatorname{SL}^{+}_{m}(\mathbf{F}_{q})$ and $\operatorname{SO}_{2m-1}(\mathbf{F}_{q})$ of rank $m-1$ with root systems $\Sigma=A_{m-1},B_{m-1}$ are the polynomial identities

[TABLE]

The sums are indexed by all $(m_{1},\ldots,m_{k})\in\mathrm{OP}(m)$ and the identities for $A_{m-1}$ use Gaussian multinomial coefficients [20, §1.7].

Example 4.10** ($\operatorname{SL}^{-}_{m}(\mathbf{F}_{q})$).**

The two identities of Corollary 4.3 with $I=\widehat{\Pi}$ for the Steinberg group $\operatorname{SL}^{-}_{2m}(\mathbf{F}_{q})$ of rank $2m-1$ and twisted rank $m$ are

[TABLE]

and for the Steinberg group $\operatorname{SL}^{-}_{2m+1}(\mathbf{F}_{q})$ of rank $2m$ and twisted rank $m$ they are

[TABLE]

where the sums run over all $(m_{1},\ldots,m_{k})\in\mathrm{OP}(m)$. These identities are obtained by analyzing the $C_{2}$-subsystems of the $C_{2}$-root system $A_{m-1}$ [4, 13.3.8]. Write $S(A_{m-1})$ for the multiset of all $C_{2}$-subsystems of $A_{m-1}$. One subsystem of $A_{2m-1}$ is $a_{2m-1}$ defined to be the free part of $A_{2m-1}$, i.e. the subsystem obtained by deleting the middle root $\alpha_{m}$. The fundamental roots of the $C_{2}$-root systems $a_{1},a_{3},a_{5},a_{7}$ are

[TABLE]

The first multisets of subsystems are $S(A_{1})=\{a_{1},A_{1}\}$, $S(A_{2})=\{a_{1},A_{2}\}$, $S(A_{3})=\{a_{1},A_{1},a_{3},A_{3}\}=a_{1}\times S(A_{1})\cup\{a_{3},A_{3}\}$, $S(A_{4})=\{a_{1},A_{2},a_{3},A_{4}\}=a_{1}\times S(A_{2})\cup\{a_{3},A_{4}\}$. In general, the $2^{m}$ subsystems of $A_{2m-1}$ and $A_{2m}$, $m\geq 2$, are the multisets

[TABLE]

For each subsystem $a$ of $A_{m}$, let $P(a)(q)=|P\mathrel{\mathop{\mathchar 58\relax}}B|\in\mathbf{Z}[q]$ be the index of the Borel subgroup $B$ in the parabolic subgroup of $\operatorname{SL}^{-}_{m+1}(\mathbf{F}_{q})$ corresponding to $a$. In particular, $P(A_{m})(q)$ and $P(a_{2m-1})(q)$ are the polynomials

[TABLE]

of degrees $\binom{m+1}{2}$ and $\binom{m}{2}$. Consider the multiset of signed polynomials associated to all subsystems of $A_{m}$

[TABLE]

where $\Pi(a)$ is the set of fundamental roots and $\Pi(a)/C_{2}$ the orbit set. Then $P(S(A_{1}))=\{1,-P(A_{1})\}$, $P(S(A_{2}))=\{1,-P(A_{2})\}$ and one may now determine the multisets of polynomials for all the $C_{2}$-root systems $A_{2m-1}$ and $A_{2m}$, $m\geq 2$. This leads to the above polynomial identities.

Steinberg’s theorem applies in the equicharacteristic case and does not hold in the cross-characteristic case. However, it is known that the number of $p$-singular classes in $\operatorname{GL}_{n}(\mathbf{F}_{q})$, $p\nmid q$, is

[TABLE]

where $\lambda$ ranges over all partitions of $n$ and $T(\lambda)$ is the number of permutations of cycle type $\lambda$ in $\Sigma_{n}$ [15, Corollary 4.22]. The number of $p$-singular classes, but not the number of $p$-singular elements, in $\operatorname{GL}_{n}(\mathbf{F}_{q})$, $p\nmid q$, depends only on the $p$-fusion system.

Acknowledgments

I thank the Department of Mathematics at the Universitat Autònoma Barcelona for kind hospitality and for the opportunity to speak in the Topology Seminar. Lemma 2.1 was pointed out by Sune Precht Ree. Warm thanks also go to Bob Oliver for his interest and Justin Lynd for referring me to Steinberg’s paper [22]. This note would not have been written without their help and encouragement.