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 [Reference Moretó3] 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 [Reference Keller and Yang1] 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 [Reference Yang6]. The main result of [Reference Keller and Yang1] 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 [Reference Keller and Yang1], it seems that the bound
$49$
is far from the best possible. For example, it was proved in [Reference Keller and Yang1] 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 [Reference Manz and Wolf2, 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 [Reference Manz and Wolf2, 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 [Reference Manz and Wolf2, 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 [Reference Keller and Yang1, 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 [Reference Yang5, 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 [Reference Qian and Yang4, 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
$2^{\mu (G)}\cdot \mathrm {Ord}_S(A) \leq b^{27.41}$
.
Acknowledgement
The authors are grateful to the referee for the valuable suggestions which improved the manuscript.
