A short note on the noncoprime regular module problem

TL;DR

This paper investigates the existence of regular modules in a specific group action setting involving finite groups and automorphisms, contributing to the understanding of module structures under group actions.

Contribution

It introduces a new perspective on the regular module problem for finite groups with automorphism actions, focusing on a particular configuration.

Findings

Identifies conditions for the existence of regular A-modules in the given setting.

Provides theoretical insights into automorphism actions on finite groups and modules.

Clarifies the structure of semidirect products in the context of module theory.

Abstract

We consider a special configuration in which a finite group $A$ acts by automorphisms on the finite group $G$, and the semidirect product $GA$ acts on the vector space $V$ by linear transformations; and discuss the existence of the regular $A$-module in $V_{_{A}}$.

Equations

$$C_A(v)=\bigcap _{i=1}^k{C}_A(w_i)=\bigcap _{i=1}^k{C}_A(W_i)=C_A(V)=1.$$

$$V\cong (X\otimes R_{j_1})\oplus \cdots \oplus (X\otimes R_{j_m})\cong X\otimes (\underset{i=1}{\overset{m}{⨁}}{R}_{j_i})$$

$$R_{j_i}=U_1\supset U_2\supset \cdots \supset U_s=0$$

Full text

*∗*Corresponding author

This work has been supported by the Research Project TÜBİTAK 114F223.

1. Introduction

Let $A$ be a finite group which acts faithfully on the vector space $V$ by linear transformations. We say “$A$ has a regular orbit on $V$” if there is a vector $v$ in $V$ such that $C_{A}(v)=1$. In this case, the $A$-orbit containing $v$ is called a regular $A$-orbit. Furthermore, $V$ contains the regular $A$-module if a regular $A$-orbit happens to be linearly independent. More generally if $A$ acts by linear transformations on the vector space $V$ (not necessarily faithfully), then we say that $A$ has a regular orbit on $V$ or $V$ contains the regular $A$-module if $A/C_{A}(V)$ does the same.

While studying the structure of a finite solvable group $G$ admitting a certain group of automorphisms $A$, we are often forced to study $A$-invariant chief factors $V$ of $G$ together with the action of the semidirect product $(G/C_{G}(V))A$ on $V.$ It turns out to be rather efficient to know that $V$ contains the regular $A$-module or at least a regular $A$-orbit. Not all groups act with regular orbits although many interesting and rich classes do, especially under the additional assumptions of coprimeness that $(|G|,|A|)=1=(|V|,|GA|).$ There has been extensive research about the existence of regular orbits such as [1], [6], [7], [8], [11], [12] in the case of coprimeness and [2], [4], [5], [13], [14] in the noncoprime case. All the results concerning a nilpotent $A$ are culminating in Theorem 1.1 in [14] which can be reformulated as follows:

*Let $G$ be a finite solvable group admitting a nilpotent group $A$ as a group of automorphisms. Suppose that $C_{O_{p}(A)}(G)=1$. Let $V$ be a finite faithful $kGA$-module over a field $k$ of characteristic $p$ not dividing the order of $G$. Then $A$ has at least one regular orbit on $V$ if $A$ involves no wreath product $\mathbb{Z}_{2}\wr\mathbb{Z}_{2}$ and involves no wreath product $\mathbb{Z}_{r}\wr\mathbb{Z}_{r}$ for $r$ a Mersenne prime when $p=2.$

In the present paper we prove a theorem which concludes the existence of a regular module without the coprimeness condition the prototype of which is Theorem 1.5 in [11]. This theorem was improved as Theorem B in [5] in case where the group $GA$ is of odd order. For the convenience of the reader, we formulate the main conclusion of Theorem 1.5 in a way suitable to emphasize the similarities and differences between this theorem and Theorem B in [5] and our result.

*Let $PRA$ be a finite group where $P$ is a $p$-group and $R$ is an $r$-group for distinct primes $p$ and $r$ not dividing the order of $A$ such that $P\lhd PRA$ and $R\lhd RA$. Assume that the following are satisfied:

(a) $P$ is an extraspecial $p$-group for some prime $p$ where $Z(P)\leqslant Z(PRA)$ and $C_{A}(P)=1$;

(b) $\bar{R}=R/R_{0}$ is of class at most two and of exponent $r$ where $R_{0}=C_{R}(P)$. Suppose that $|C_{A}(\bar{R}/\Phi(\bar{R})|$ is either a prime or $1$;

(c) $A/C_{A}(\bar{R}/\Phi(\bar{R})$ has a regular orbit in its action on $\bar{R}/\Phi(\bar{R})$;

if $C_{A}(\bar{R}/\Phi(\bar{R})\neq 1,$ $[C_{A}(\bar{R}/\Phi(\bar{R}),P]\neq P$ and $p=2$, assume that $|C_{A}(\bar{R}/\Phi(\bar{R})|$ is not a Fermat prime.

Let $\chi$ be a complex $PRA$-character such that $\chi_{{}_{P}}$ is faithful. Then $\chi_{{}_{A}}$ contains the regular $A$-character.

Namely we obtain the following:

Theorem *Let $PRA$ be a finite group where $P$ is a $p$-group and $R$ is an $r$-group for distinct primes $p$ and $r$ such that $P\lhd PRA$ and $R\lhd RA$. Assume that the following are satisfied:

(a) $P$ is an extraspecial $p$-group for some prime $p$ where $Z(P)\leqslant Z(PRA)$ and $C_{A}(P)=1$;

(b) $R/R_{0}$ is of class at most two and of exponent dividing $r$ where $R_{0}=C_{R}(P)$ and $A_{0}=C_{A}(R/R_{0})=1;$

(c) $A=A_{p}\times A_{r}\times A_{\{p,r\}^{\prime}}$ where its Sylow $r$-subgroup $A_{r}$ and Sylow $p$-subgroup $A_{p}$ are both cyclic and $A_{\{p,r\}^{\prime}}$ acts with regular orbits on $R/\Phi(R)$,

(d) if $p=2$ then $r$ is not a Fermat prime.

Let $\chi$ be a complex $PRA$-character such that $\chi_{{}_{P}}$ is faithful. Then $\chi_{{}_{A}}$ contains the regular $A$-character.

Notice that both $p$ and $r$ are allowed to divide the order of $A$.

All groups considered in this paper are finite and the notation and terminology are standard.

2. Existence of regular orbits

In this section we present a result due to Dade [3] on the existence of regular orbits which will be applied in the proof of our theorem.

Proposition *Let V be a faithful $kA$-module over a finite field $k$ of characteristic $p$. Assume that $A=B\times C$ where $B$ is a cyclic $p$-group and $C$ is a $p^{\prime}$-group which has a regular orbit on every $C$-invariant irreducible section of $V$. Then $A$ has a regular orbit on $V$. *

Proof.

Let $V_{C}=W_{1}\oplus\cdots\oplus W_{\ell}$ be the decomposition of $V$ into its $C$-homogenous components. As $B$ and $C$ commute, each $W_{i}$ is $A$-invariant. Therefore it suffices to prove that $A$ has a regular orbit on $W_{i}$, for each $i=1,\ldots,\ell$. To see this let $w_{i}\in W_{i}$ be such that $C_{A}(w_{i})=C_{A}(W_{i})$ for $i=1,\ldots,\ell$. If $v=w_{1}+\cdots+w_{\ell}$, then

[TABLE]

Thus we may assume that $\ell=1$, that is, $V_{C}$ is homogeneous. Let $X$ be the irreducible $kC$-module which appears in $V_{C}$ and let $B=\left\langle\alpha\right\rangle$. Then we have $kB=k[\alpha-1]$. Set $R_{j}=kB/\left\langle(\alpha-1)^{j}\right\rangle$, for $j=1,\ldots,p^{n}$ where $p^{n}=|\alpha|$. Note that $R_{j}$ is an indecomposable $kB$-module of dimension $j$ for each $j$ and these are the only indecomposable $kB$-modules by Theorem VII.5.3 in [9]. Then the decomposition of the $kA$-module $V$ into indecomposable $kA$-modules can be given as

[TABLE]

for some $j_{1},\ldots,j_{m}$ in $\{1,\ldots,p^{n}\}$. To simplify the notation we set $U=\bigoplus_{i=1}^{m}R_{j_{i}}$. The group $C$ has a regular orbit on $X$ by hypothesis, that is, there is $x\in X$ such that $C_{C}(x)=C_{C}(X)=1$. We shall observe that $B$ has a regular orbit on $U$ : As a consequence of the faithful action of $A$ on $V$, $B$ acts faithfully on $U$. Hence there is at least one indecomposable component, say $R_{j_{i}}$, on which $B$ acts faithfully, since $B$ is cyclic. Let

[TABLE]

be a $B$-composition series of $R_{j_{i}}=U_{1}$. Each factor $U_{i}/U_{i+1}$, $i=1,\ldots,s-1$, is isomorphic to the trivial module of dimension $1$. Hence $s-1=dimU_{1}=j_{1}$ and $[U_{1},\underbrace{\alpha,\ldots,\alpha}_{j_{1}-times}]=0$. It follows that $dimU_{1}=j_{1}\geqslant p^{n-1}+1$, because otherwise $(\alpha-1)^{p^{n-1}}=0$ on $U_{1}$ and hence $\alpha^{p^{n-1}}$ is trivial on $U_{1}$, a contradiction. Pick an element $u$ from $U_{1}-U_{2}$. If $C_{B}(u)\neq 1$, then $\alpha^{p^{n-1}}$ acts trivially on $u$, whence the degree $j_{1}$ of the minimum polynomial of $\alpha$ on $U_{1}$ is at most $p^{n-1}$ . But then $p^{n-1}+1\leqslant j_{1}\leqslant p^{n-1}$, which is impossible. This yields that $C_{B}(u)=1=C_{B}(U)$. As a consequence, $B$ has a regular orbit on $U$. We are now ready to complete the proof of the theorem. Let $a\in C_{A}(x\otimes u)$. Then $a=b+c$ for some $b\in B$ and $c\in C$. As $c\in\left\langle a\right\rangle$, we have $(x\otimes u)c=xc\otimes u=x\otimes u$ and hence $xc=x$. That is, $c\in C_{C}(x)=C_{C}(X)$. Similarly, we observe that $b\in C_{B}(u)=C_{A}(U)$. Consequently, we have $a\in C_{A}(X\otimes U)$ and hence the equality $C_{A}(x\otimes u)=C_{A}(X\otimes U)$ holds. It follows that $A$ has regular orbit on $V$, as claimed. ∎

Remark 2.1*.*

The above proposition cannot be extended to abelian $O_{p}(A)$ as the following example shows: Let $V$ be an elementary abelian group of order $p^{3}$ with a basis $\{v_{1},v_{2},v_{3}\}$ and $A$ an elementary abelian group of order $p^{2}$ of automorphisms of $V$ generated by $\{a_{1},a_{2}\}$ with the action $v_{1}^{a_{1}}=v_{1}^{a_{2}}=v_{1},v_{2}^{a_{1}}=v_{1}v_{2},v_{2}^{a_{2}}=v_{2},v_{3}^{a_{1}}=v_{3},v_{3}^{a_{2}}=v_{3}v_{1}.$ Then every $A$-orbit on $V$ has length dividing $p.$

3. Proof of theorem

Let $(P,R,\chi)$ be a counterexample with $\left|PR\right|+\chi(1)$ minimum. We shall proceed in a series of steps. To simplify the notation we set $G=PR$ .

*(1) $\chi$ is irreducible.

There exists an irreducible constituent $\chi_{{}_{1}}$ of $\chi$ which does not contain $Z(P)$ in its kernel, that is $(\chi_{{}_{1}})_{{}_{P}}$ is faithful. Then we have $\chi_{{}_{1}}=\chi$ because otherwise $\chi_{{}_{1}}$ contains the regular $A$-character by induction.

*(2) $\chi_{{}_{P}}$ is homogeneous and $R_{0}=1.$

As it is well known the irreducible characters of the extraspecial group $P$ are uniquely determined by their restriction $Z(P)$ so that $\chi_{{}_{P}}=e\theta$ for some faithful irreducible $GA$-invariant character $\theta$ of $P$ and some positive integer $e$, since $Z(P)\leqslant Z(GA)$. The coprimeness condition $(|P|,|RA_{p^{\prime}}|)=1$ enables us to extend $\theta$ in a unique way to an irreducible character $\overline{\theta}$ of $GA_{p^{\prime}}$ such that $det(\overline{\theta})(x)=1$ for each $x\in RA_{p^{\prime}}$ by [[10], 8.16]. On the other hand $\theta_{1}=\theta\times 1_{R_{0}}$ is an irreducible $P\times R_{0}$-character with $R_{0}\leqslant Ker\theta_{1}$. We can extend $\theta_{1}$ uniquely to $\overline{\theta}_{1}\in Irr(GA_{p^{\prime}}/R_{0})$ with $det(\overline{\theta}_{1})(x)=1$ for each $x\in RA_{p^{\prime}}/R_{0}$. The uniqueness of this extension implies $R_{0}\leqslant Ker\overline{\theta}$. Notice that $(\overline{\theta}_{1})_{{}_{P}}=\theta=\overline{\theta}_{{}_{P}}$ and also that the set $\{\varphi\,:\,\varphi\in Irr(GA_{p^{\prime}})\,\,\text{such that}\,\varphi_{{}_{P}}=\theta\}$ is $A_{p}$-invariant, because $\theta^{a}=\theta$ for each $a\in A_{p}$. Since $det(\overline{\theta}^{a})(x)=1$ for each $a\in A_{p}$, the uniqueness of $\overline{\theta}$ gives $\overline{\theta}^{a}=\overline{\theta}$. It follows from [[10], Corollary 11.22] that $\overline{\theta}$ is extendible to an irreducible $GA$-character, say $\overline{\overline{\theta}}$ . Now $\overline{\overline{\theta}}_{{}_{G}}=\overline{\theta}$, $\overline{\overline{\theta}}_{{}_{P}}=\theta$ and $R_{0}\leqslant Ker\overline{\theta}=G\cap Ker\overline{\overline{\theta}}$. If $\overline{\overline{\theta}}(1)<\chi_{{}_{1}}$ or $R_{0}\neq 1$, by induction applied to the group $GA/R_{0}$ over $\overline{\overline{\theta}}$ we see that $\overline{\overline{\theta}}_{{}_{A}}$ contains the regular $A$ -character. Since $\chi$ is a constituent of $\overline{\overline{\theta}}_{{}_{P}}|^{GA}$, there exists $\beta\in Irr(GA/P)$ such that $\chi=\overline{\overline{\theta}}\cdot{\beta}$ by [[10], 6.17] and hence $\chi_{{}_{A}}=\overline{\overline{\theta}}_{{}_{A}}\cdot\beta_{A}$. We conclude that $\chi_{{}_{A}}$ contains the regular $A$-character, while $\overline{\overline{\theta}}_{{}_{A}}$ does. Therefore without loss of generality we may assume that $R_{0}=1$ as claimed.

*(3) Theorem follows.

Theorem 1.3 in [11] applied to the group $PR$ over $\chi$ shows that one of the following holds:

$(i)$ $\chi_{{}_{{R}}}$ contains the regular $R$-character;

$(ii)$ $p=2$, and $r$ is a Fermat prime.

By hypothesis $(d)$ we see that $(i)$ follows, that is $\chi_{{}_{R}}$ contains a copy of every irreducible $R$-character. On the other hand we can regard $Irr(R/\Phi(R))$ as a faithful $\mathbb{F}_{r}(A)$-module which is isomorphic to $R/\Phi(R)$ and hence apply the proposition above to get a linear character $\nu$ of $R$ such that $C_{A}(\nu)=1$. Let $V$ be a $GA$-module affording $\chi$ and let $W$ be the homogeneous component of $V_{{}_{R}}$ corresponding to $\nu$. Since the stabilizer in $A$ of $W$ is trivial, $V_{{}_{A}}$ contains the regular $A$-module. Therefore $\chi_{{}_{A}}$ contains the regular $A$-character. $\Box$