ON LARGE ORBITS OF FINITE SOLVABLE GROUPS ON CHARACTERS

Abstract

We prove that if a solvable group A acts coprimely on a solvable group G, then A has a relatively ‘large’ orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This improves an earlier result of Keller and Yang [‘Orbits of finite solvable groups on characters’, Israel J. Math. 199 (2014), 933–940].

1 Introduction

Let a finite group A act (via automorphisms) on a finite group G. Such an action induces an action of A on the set ${\operatorname {Irr}}(G)$ in an obvious way (where ${\operatorname {Irr}}(G)$ denotes the set of complex irreducible characters of G). When G is elementary abelian, we are back to studying linear group actions. However, for nonabelian G, not much is known about this interesting action and we are only aware of a few major results on the action of A on ${\operatorname {Irr}}(G)$ .

One such result is due to Moretó [3] who proved the existence of a ‘large’ orbit on ${\operatorname {Irr}}(G)$ when A is a p-group for some prime p and G is solvable such that $(|A|,|G|)=1$ . Keller and Yang [1] extended this result and established the existence of a ‘large’ orbit on ${\operatorname {Irr}}(G)$ whenever both A and G are solvable with $(|A|,|G|)=1$ . Yang also studied the special situation where A is nilpotent in [6]. The main result of [1] is the following theorem.

Theorem 1.1. Let A and G be finite solvable groups such that A acts faithfully and coprimely on G. Let b be an integer such that $|A : {\mathbf {C}}_A(\hspace{2pt}\chi )| \leq b$ for all $\chi \in {\operatorname {Irr}}(G)$ . Then $|A| \leq b^{49}$ .

As discussed in [1], it seems that the bound $49$ is far from the best possible. For example, it was proved in [1] that if $2, 3\notin \pi =\pi (A)$ , then $|A|\leq b^4$ . It was also remarked that the best bound is probably close to $b^2$ . It would be interesting to construct nontrivial examples in GAP but this seems challenging.

The main purpose of this note is to provide a modest improvement on the bound. The main idea is to restructure the group decomposition and estimate the bound from a different perspective. We prove the following result.

Theorem 1.2. Let A and G be finite solvable groups such that A acts faithfully and coprimely on G. Let b be an integer such that $|A : {\mathbf {C}}_A(\hspace{2pt}\chi )| \leq b$ for all $\chi \in {\operatorname {Irr}}(G)$ . Then $|A| \leq b^{27.41}$ .

2 Notation and preliminary results

We first fix some notation. In this paper, we use ${\mathbf {F}}(G)$ to denote the Fitting subgroup of G. Let ${\mathbf {F}}_0(G) \leq {\mathbf {F}}_1(G) \leq {\mathbf {F}}_2(G) \leq \cdots \leq {\mathbf {F}}_n(G)=G$ denote the ascending Fitting series, that is, ${\mathbf {F}}_0(G)=1$ , ${\mathbf {F}}_1(G)={\mathbf {F}}(G)$ and ${\mathbf {F}}_{i+1}(G)/{\mathbf {F}}_i(G)={\mathbf {F}}(G/{\mathbf {F}}_i(G))$ . Here, ${\mathbf {F}}_i(G)$ is the ith ascending Fitting subgroup of G. We use ${\operatorname {fl}}(G)$ to denote the Fitting length of the group G. We use $\Phi (G)$ to denote the Frattini subgroup of G.

Proposition 2.1 [2, Theorem 3.5(a)].

Let G be a finite solvable group and let ${V \neq 0}$ be a finite, faithful, completely reducible G-module. Then $|G| \leq |V|^\alpha / \lambda $ , where $\alpha= {\ln ((24)^{1/3} \cdot 48)}/{\ln 9}$ and $\lambda =24^{1/3}$ .

Proposition 2.2. Let G be a finite solvable group and let $V \neq 0$ be a finite, faithful, completely reducible G-module. Suppose ${\operatorname {fl}}(G)\leq 2$ . Then $|G| \leq |V|^\gamma / \eta $ , where $\gamma = {\ln ((6)^{1/2} \cdot 24)}/{\ln 9}$ and $\eta =6^{1/2}$ .

Proof. One can mimic the proof of [2, Theorem 3.5(a)]. Note that one has to avoid $S_4$ and ${\operatorname {GL}}(2,3)$ in the group structure since ${\operatorname {fl}}(S_4)=3$ and ${\operatorname {fl}}({\operatorname {GL}}(2,3))=3$ .

Proposition 2.3 [2, Theorem 3.3(a)].

Let G be a finite nilpotent group and let ${V \neq 0}$ be a finite, faithful, completely reducible G-module. Then $|G| \leq |V|^\beta / 2$ , where $\beta= {\ln 32}/{\ln 9}$ .

Proposition 2.4 [1, Theorem 3.1].

Assume that a solvable $\pi $ -group A acts faithfully on a solvable $\pi '$ -group G. Let b be an integer such that $|A : {\mathbf {C}}_A(\hspace{2pt}\chi )| \leq b$ for all ${\chi \in {\operatorname {Irr}}(G)}$ . Let $\Gamma = AG$ be the semidirect product. Let $K_{i+1}={\mathbf {F}}_{i+1}(\Gamma )/{\mathbf {F}}_i(\Gamma )$ and let $K_{i+1, \pi }$ be the Hall $\pi $ -subgroup of $K_{i+1}$ for all $i \geq 1$ . Let $K_i/\Phi (\Gamma /{\mathbf {F}}_{i-1}(\Gamma ))=V_{i1}+V_{i2}$ , where $V_{i1}$ is the $\pi $ part of $K_i/\Phi (\Gamma /{\mathbf {F}}_{i-1}(\Gamma ))$ and $V_{i2}$ is the $\pi '$ part of $K_i/\Phi (\Gamma /{\mathbf {F}}_{i-1}(\Gamma ))$ for all $i \geq 1$ . Let $K \triangleleft \Gamma $ such that ${\mathbf {F}}_i(\Gamma ) \triangleleft K$ . Let $L_{i+1, \pi }=K_{i+1, \pi } \cap K$ . Then $|{\mathbf {C}}_{L_{i+1, \pi }}(V_{i1})| \leq b^2$ and $|{\mathbf {C}}_{L_{i+1, \pi }}(V_{i1})| \leq b$ if $L_{i+1, \pi }$ is abelian. The order of the maximum abelian quotient of ${\mathbf {C}}_{L_{i+1, \pi }}(V_{i1})$ is less than or equal to b for all $i \geq 1$ .

3 Main results

Now we are ready to prove Theorem 1.2, which we restate here.

Theorem 3.1. Let A be a solvable $\pi $ -group that acts faithfully on a solvable $\pi '$ -group G. Let b be an integer such that $|A : {\mathbf {C}}_A(\chi )| \leq b$ for all $\chi \in {\operatorname {Irr}}(G)$ . Then $|A| \leq b^{27.41}$ .

Proof. Let $\Gamma = AG$ be the semidirect product of A and G. By Gaschutz’s theorem, $\Gamma /{\mathbf {F}}(\Gamma )$ acts faithfully and completely reducibly on ${\operatorname {Irr}}({\mathbf {F}}(\Gamma )/\Phi (\Gamma ))$ . It follows from [5, Theorem 3.3] that there exists $\lambda \in {\operatorname {Irr}}({\mathbf {F}}(\Gamma )/\Phi (\Gamma ))$ such that $T = {\mathbf {C}}_{\Gamma }(\lambda ) \leq {\mathbf {F}}_8(\Gamma )$ .

Let $K_2={\mathbf {F}}_2(\Gamma )/{\mathbf {F}}_1(\Gamma )$ and let $K_{2, \pi }$ be the Hall $\pi $ -subgroup of $K_2$ . Then $K_{2, \pi }$ acts faithfully and completely reducibly on $K_1/\Phi (\Gamma /{\mathbf {F}}_{0}(\Gamma ))$ . It is clear that we may write $K_1/\Phi (\Gamma /{\mathbf {F}}_{0}(\Gamma ))=V_{11}+V_{12}$ , where $V_{11}$ is the $\pi $ part of $K_1/\Phi (\Gamma /{\mathbf {F}}_{0}(\Gamma ))$ and $V_{12}$ is the $\pi '$ part of $K_1/\Phi (\Gamma /{\mathbf {F}}_{0}(\Gamma ))$ .

It is also clear that $K_1={\mathbf {F}}(\Gamma )$ is a $\pi '$ -group and $V_{11}=0$ . Thus, $K_{2,\pi }={\mathbf {C}}_{K_{2, \pi }}(V_{11})$ acts faithfully and completely reducibly on $V_{12}$ . Proposition 2.4 shows that $|K_{2,\pi }| \leq b^2$ and the order of the maximum abelian quotient of $K_{2,\pi }$ is bounded above by b (and thus $|V_{22}| \leq b)$ .

Set $G_2{\kern-1pt}={\kern-1pt}{\mathbf {F}}_8(\Gamma )/{\mathbf {F}}(\Gamma )$ and $G_3{\kern-1pt}={\kern-1pt}{\mathbf {C}}_{G_2/{\mathbf {F}}(G_2)}(V_{21})$ . Thus, $|G_2/{\mathbf {F}}(G_2)/{\mathbf {C}}_{G_2/{\mathbf {F}}(G_2)}(V_{21})| {\kern-1pt}\leq{\kern-1pt} b^{\alpha}$ by Proposition 2.1. We note that $G_3$ acts faithfully and completely reducibly on $V_{22}$ and ${\operatorname {fl}}(G_3) \leq 6$ .

Let ${\mathbf {F}}(G_3)/\Phi (G_3)=V_{31}+V_{32}$ , where $V_{31}$ is the $\pi $ part of ${\mathbf {F}}(G_3)/\Phi (G_3)$ and $V_{32}$ is the $\pi '$ part of ${\mathbf {F}}(G_3)/\Phi (G_3)$ . Proposition 2.4 shows that the order of the $\pi $ part of ${\mathbf {F}}(G_3)$ is bounded by $b^2$ and the order of the abelian quotient of the $\pi $ part of ${\mathbf {F}}(G_3)$ is bounded by b (and thus $|V_{32}| \leq b)$ .

Set $G_4={\mathbf {C}}_{G_3/{\mathbf {F}}(G_3)}(V_{31})$ . Thus, $|G_3/{\mathbf {F}}(G_3)/{\mathbf {C}}_{G_3/{\mathbf {F}}(G_3)}(V_{31})| \leq b^{\alpha}$ by Proposition 2.1. We note that $G_4$ acts faithfully and completely reducibly on $V_{32}$ and ${\operatorname {fl}}(G_4) \leq 5$ .

Let ${\mathbf {F}}(G_4)/\Phi (G_4)=V_{41}+V_{42}$ , where $V_{41}$ is the $\pi $ part of ${\mathbf {F}}(G_4)/\Phi (G_4)$ and $V_{42}$ is the $\pi '$ part of ${\mathbf {F}}(G_4)/\Phi (G_4)$ . Proposition 2.4 shows that the order of the $\pi $ part of ${\mathbf {F}}(G_4)$ is bounded by $b^2$ and the order of the abelian quotient of the $\pi $ part of ${\mathbf {F}}(G_4)$ is bounded by b (and thus $|V_{42}| \leq b)$ .

Set $G_5={\mathbf {C}}_{G_4/{\mathbf {F}}(G_4)}(V_{41})$ . Thus, $|G_4/{\mathbf {F}}(G_4)/{\mathbf {C}}_{G_4/{\mathbf {F}}(G_4)}(V_{41})| \leq b^{\alpha}$ by Proposition 2.1. We note that $G_5$ acts faithfully and completely reducibly on $V_{42}$ and ${\operatorname {fl}}(G_5) \leq 4$ .

Let ${\mathbf {F}}(G_5)/\Phi (G_5)=V_{51}+V_{52}$ , where $V_{51}$ is the $\pi $ part of ${\mathbf {F}}(G_5)/\Phi (G_5)$ and $V_{52}$ is the $\pi '$ part of ${\mathbf {F}}(G_5)/\Phi (G_5)$ . Proposition 2.4 shows that the order of the $\pi $ part of ${\mathbf {F}}(G_5)$ is bounded by $b^2$ and the order of the abelian quotient of the $\pi $ part of ${\mathbf {F}}(G_5)$ is bounded by b (and thus $|V_{52}| \leq b)$ .

Set $G_6={\mathbf {C}}_{G_5/{\mathbf {F}}(G_5)}(V_{51})$ . Thus, $|G_5/{\mathbf {F}}(G_5)/{\mathbf {C}}_{G_5/{\mathbf {F}}(G_5)}(V_{51})| \leq b^{\alpha}$ by Proposition 2.1. We note that $G_6$ acts faithfully and completely reducibly on $V_{52}$ and ${\operatorname {fl}}(G_6) \leq 3$ .

Let ${\mathbf {F}}(G_6)/\Phi (G_6)=V_{61}+V_{62}$ , where $V_{61}$ is the $\pi $ part of ${\mathbf {F}}(G_6)/\Phi (G_6)$ and $V_{62}$ is the $\pi '$ part of ${\mathbf {F}}(G_6)/\Phi (G_6)$ . Proposition 2.4 shows that the order of the $\pi $ part of ${\mathbf {F}}(G_6)$ is bounded by $b^2$ and the order of the abelian quotient of the $\pi $ part of ${\mathbf {F}}(G_6)$ is bounded by b (and thus $|V_{62}| \leq b)$ .

Set $G_7={\mathbf {C}}_{G_6/{\mathbf {F}}(G_6)}(V_{61})$ . Thus, $|G_6/{\mathbf {F}}(G_6)/{\mathbf {C}}_{G_6/{\mathbf {F}}(G_6)}(V_{61})| \leq b^{\gamma}$ by Proposition 2.2. We note that $G_7$ acts faithfully and completely reducibly on $V_{62}$ and ${\operatorname {fl}}(G_7) \leq 2$ .

Let ${\mathbf {F}}(G_7)/\Phi (G_7)=V_{71}+V_{72}$ , where $V_{71}$ is the $\pi $ part of ${\mathbf {F}}(G_7)/\Phi (G_7)$ and $V_{72}$ is the $\pi '$ part of ${\mathbf {F}}(G_7)/\Phi (G_7)$ . Proposition 2.4 shows that the order of the $\pi $ part of ${\mathbf {F}}(G_7)$ is bounded by $b^2$ and the order of the abelian quotient of the $\pi $ part of ${\mathbf {F}}(G_7)$ is bounded by b (and thus $|V_{72}| \leq b)$ .

Set $G_8={\mathbf {C}}_{G_7/{\mathbf {F}}(G_7)}(V_{71})$ . Thus, $|G_7/{\mathbf {F}}(G_7)/{\mathbf {C}}_{G_7/{\mathbf {F}}(G_7)}(V_{71})| \leq b^{{\kern1.2pt}\beta}$ by Proposition 2.3. We note that $G_8$ acts faithfully and completely reducibly on $V_{72}$ and ${\operatorname {fl}}(G_8) \leq 1$ . Proposition 2.4 shows that the order of the $\pi $ part of $G_8={\mathbf {F}}(G_8)$ is bounded by $b^2$ .

Next, we show that $|\Gamma : T|_{\pi } \leq b$ .

Let $\chi $ be any irreducible character of G lying over $\lambda $ . Then every irreducible character of $\Gamma $ that lies over $\chi $ also lies over $\lambda $ and hence has degree divisible by $|\Gamma : T|$ . However, $\chi $ extends to its stabiliser in $\Gamma $ and thus some irreducible character of $\Gamma $ lying over $\chi $ has degree $\chi (1) |A : C_A(\,\chi )|$ . Therefore, the $\pi $ -part of $|\Gamma : T|$ divides $|A : {\mathbf {C}}_A(\,\chi )|$ which is at most b. This gives

$$ \begin{align*} |A| \leq b^{2 \cdot 7} \cdot b^{\alpha \cdot 4} \cdot b^{\gamma} \cdot b^{\beta} \cdot b \leq b^{27.41},\end{align*} $$

and the result follows.

When $(|A|,|G|)=1$ , the orbit sizes of A on ${\operatorname {Irr}}(G)$ are the same as the orbit sizes in the natural action of A on the conjugacy classes of G. The following result follows immediately from Theorem 1.2.

Theorem 3.2. Let A be a solvable $\pi $ -group that acts faithfully on a solvable $\pi '$ -group G. Let b be an integer such that $|A : {\mathbf {C}}_A(C)| \leq b$ for all $C \in {\operatorname {cl}}(G)$ . Then $|A| \leq b^{27.41}$ .

We now give an application of our main result. Take a chief series

$$ \begin{align*}\Delta: 1=G_0< G_1< \cdots <G_n=G\end{align*} $$

of a finite group G. Let $\mathrm {Ord}_{\mathcal {S}}(G)$ denote the product of the orders of all solvable chief factors $G_i/G_{i-1}$ in $\Delta $ . Let $\mu (G)$ be the number of nonabelian chief factors in $\Delta $ . Clearly, the constants $\mathrm { Ord}_{\mathcal {S}}(G)$ and $\mu (G)$ are independent of the choice of chief series $\Delta $ of G. As an application of Theorem 3.1, we can strengthen the solvable case of [4, Theorem 4.7].

Theorem 3.3. Let a finite group A act faithfully on a finite group G with $(|A|, |G|)=1$ . Assume G is solvable. If b is an integer such that $|A : {\mathbf {C}}_A(\hspace{2pt}\chi )| \leq b$ for all $\chi \in {\operatorname {Irr}}(G)$ , then $2^{\mu (G)}\cdot \mathrm {Ord}_S(A) \leq b^{27.41}$ .