Large orbit sizes in finite group actions

Abstract

In this paper, we study the relations of the sizes of various sections of finite linear groups and the largest orbit size of the linear group actions. We also study various applications of those orbit theorems.

Introduction

In this paper, G always denotes a finite group. It is a classical theme to study the G-orbit structure of V where G acts faithfully on a finite vector space V. Here the orbit of an element $v\in V$ is the set of elements in V to which v can be moved by the elements of G. One of the most important and natural questions is to study the existence of an orbit of a certain size. Results on orbit sizes, particularly on the existence of large orbits, have been fundamental to solving problems in the modular and ordinary character theory of finite groups. For a long time, there has been a deep interest in examining the size of the largest possible orbits in group actions. Let v be an element in V, an orbit $\{v^g|g\in G\}$ is called regular, if $C_G(v)=1$ holds or equivalently the size of the orbit $v^G$ is $|G|$. For the existence of regular orbits of primitive solvable linear groups, please see some former results of the second author in [22], [23], [24].

Although the existence of regular orbit is not guaranteed in general even for solvable linear groups, researchers have found various alternative ways to measure the size of the largest orbit. One approach is to show the existence of regular orbits of G on $V\oplus V$ or $V\oplus V\oplus V$. Observe that if G has a regular orbit on $V\oplus V$, then G has a large orbit on V of size at least $\sqrt{|G|}$.

For a finite group G, let $\mathrm{b}(G)$ denote the largest irreducible complex character degree of G, and let $\mathbf{F}(G)$ denote the Fitting subgroup of G. For G solvable, Gluck [9] showed that $|G:\mathbf{F}(G)|\le \mathrm{b}{(G)}^{13/2}$, and conjectured that $|G:\mathbf{F}(G)|\le \mathrm{b}{(G)}^2$. While studying this conjecture, Espuelas [8] proved that if V is a finite, faithful and completely reducible G-module of odd characteristic and G is of odd order, then G has at least one regular orbit on $V\oplus V$. Dolfi and Jabara [7, Theorem 1.3] generalized Espuelas' result to the case where the Sylow 2-subgroups of GV are abelian. A similar result [24, Theorem 2.3] about large orbit size by the second author showed the following: If V is a faithful and completely reducible G-module for a solvable group G with $3\nmid |G|$, then G has at least three regular orbits on $V\oplus V$. For arbitrary solvable linear groups, Moretó and Wolf [19] showed that G has a regular orbit on $V\oplus V\oplus V$, which implies the following theorem.

Theorem 1.1

([19, Corollary 2.6]) Let G be a solvable group and V be a finite, faithful and completely reducible G-module. Then G has an orbit on V of size at least $|G{|}^{1/3}$.

Now let us consider the case when V is a finite faithful and completely reducible G-module, possibly of mixed characteristic (that is, V is a direct sum of some irreducible G-modules $V_i$, where the fields of definition of $V_i$ may be different). It is known for a long time that if G is abelian, then G has a regular orbit on V, and that if G is nilpotent, then G has a large orbit on V of size at least $\sqrt{|G|}$ (cf. [12]). Let $\mathcal{A},\mathcal{N},\mathcal{S}$ denote the class of abelian groups, the class of nilpotent groups and the class of solvable groups, respectively. For every class $\mathcal{F}\in \{\mathcal{A},\mathcal{N},\mathcal{S}\}$, denote by $G^{\mathcal{F}}$ the smallest normal subgroup of G such that $G/G^{\mathcal{F}}\in \mathcal{F}$. Note that $G^{\mathcal{A}}=G^{\prime }$. For solvable groups, Keller and the second author [15] proved the following results.

Theorem 1.2

([15, Theorem 1.1 and Theorem 3.5]) Let G be a finite solvable group and V be a finite faithful and completely reducible G-module, possibly of mixed characteristic. Let M be the largest orbit size in the action of G on V. Then

(1) $M\ge |G/G^{\prime }|$;

(2) $M\ge \sqrt{|G/G^{\mathcal{N}}|}$.

In this paper, we will generalize Theorems 1.1 and Theorem 1.2 to arbitrary finite groups. Note that a generalization of Theorem 1.2(1) to arbitrary finite groups has been obtained in [16]. However, the proof presented there seems more complicated than needed.

In this paper, we will study the relations of the abelian, nilpotent, and solvable quotients with the largest orbit size. Inspired by the results in [11], we note that the results of nilpotent quotients and solvable quotients could be further strengthened to something much stronger, and we now define some terminologies.

Given a chief series

$$\mathrm{\Delta }:1=G_0<G_1<\cdots G_n=G$$

of G. Let ${\mathrm{Ord}}_{\mathcal{S}}(G)$ and ${\mathrm{Ord}}_{\mathcal{N}}(G)$ denote the product of orders of all solvable chief factors $G_i/G_{i-1}$, and the product of orders of all central chief factors $G_j/G_{j-1}$ (that is, $[G,G_j]\le G_{j-1}$), respectively in Δ. We also write ${\mathrm{Ord}}_{\mathcal{A}}(G)=|G/G^{\prime }|$.

Let $\mu (G)$ be the number of nonabelian chief factors in Δ. Clearly, by the Jordan-Hölder theorem, the constants ${\mathrm{Ord}}_{\mathcal{S}}(G)$, ${\mathrm{Ord}}_{\mathcal{N}}(G)$ and $\mu (G)$ are independent of the choice of chief series Δ of G. For more results related to ${\mathrm{Ord}}_{\mathcal{S}}(G)$ and ${\mathrm{Ord}}_{\mathcal{N}}(G)$, we refer the readers to [11]. In this paper, we always write

$$\delta (\mathcal{A})=1,\delta (\mathcal{N})=\frac{1}{2},\delta (\mathcal{S})=\frac{1}{3}.$$

Our main results are as follows.

Theorem 1.3

Let a finite group G act faithfully on a finite group V, and M be the largest orbit size in the action of G on V. Then any one of the following conditions guarantees that

$$M\ge {(2^{\mu (G)}\cdot {\mathrm{Ord}}_{\mathcal{F}}(G))}^{\delta (\mathcal{F})}$$

for every $\mathcal{F}\in \{\mathcal{A},\mathcal{N},\mathcal{S}\}$,

(1) V is a p-group and $O_p(G)=1$ for some prime p;

(2) V is a completely reducible G-module, possibly of mixed characteristic. □

We remark here that for $\mathcal{F}=\mathcal{A}$, the previous result confirms [15, Conjecture 2.4 and Conjecture 2.5].

Our approach relies on the following key proposition. Instead of trying to prove our results directly, we obtain a reduction result that will reduce all the cases to solvable groups.

Proposition 1.4

Let G be a finite group with $O_p(G)=1$ for some prime p, and let $\mathcal{F}\in \{\mathcal{A},\mathcal{N},\mathcal{S}\}$. Then there exists a solvable subgroup H of G with $O_p(H)=1$ such that ${\mathrm{Ord}}_{\mathcal{F}}(H)\ge 2^{\mu (G)}\cdot {\mathrm{Ord}}_{\mathcal{F}}(G)$.

We shall mention that this proof technique can be adjusted to study many other related questions. For example, we show the following corollaries.

Theorem 1.5

Let G be a permutation group of degree n. Then

(1) $2^{\mu (G)}\cdot {\mathrm{Ord}}_{\mathcal{N}}(G)\le 2^{n-1}$,

(2) $2^{\mu (G)}\cdot {\mathrm{Ord}}_{\mathcal{S}}(G)\le {24}^{\frac{n-1}{3}}$.

We note that Theorem 1.5(2) strengthens [6, Theorem 5.8B].

Theorem 1.6

([1, Theorem 3]) Let V be a finite dimensional vector space over a finite field of characteristic p and G a subgroup of $\mathrm{GL}(V)$ with $O_p(G)=1$. Then $|G:G^{\prime }|<|V|$.

The following result is an application of Theorem 1.3, and can be compared with [21, Theorem 3.1 and 3.2].

Theorem 1.7

Let G be a finite group. Then

$$\mathrm{b}(G)\ge (2^{\mu (G)}\cdot {\mathrm{Ord}}_{\mathcal{F}}{(G/\mathbf{F}(G))}^{\delta (\mathcal{F})}$$

for every $\mathcal{F}\in \{\mathcal{A},\mathcal{N},\mathcal{S}\}$.

The paper is organized as follows. In sections 2 and 3, we will prove Theorem 1.3 and Proposition 1.4. In section 4, we will see some applications including Theorem 1.5, Theorem 1.7.