Finite groups all of whose proper subgroups have few character values

1.   Introduction

We always think that groups under consideration are all finite. Let $G$ be a group and $\mathrm{Irr}(G)$ be the set of all complex irreducible characters of a group $G$. Let $g$ be an element of a group $G$. Then denote by $\mathrm{cv}(G) = \{\chi(g):\chi\in \mathrm{Irr}(G), g\in G\}$, the set of character values of $G$, so $\mathrm{cd}(G)\subseteq \mathrm{cv}(G)$ where $\mathrm{cd}(G) = \{\chi(1):\chi\in \mathrm{Irr}(G)\}$ is the set of character degrees of a group $G$. We will use these symbols in this paper:

$E_{p^n}$: the elementary abelian $p$-group of order $p^n$;

$C_n$: the cyclic group of order $n$;

$Q_8$: the quaternion group of order $8$;

$D_{2n}$: the dihedral group of order $2n$.

Character values of finite groups have large influence on their structures; see ^[1,2,23,26]. Recently, Madanha ^[17] and Sakural ^[24] studied the influence of character values in a character table on the structure of a group respectively. In particular, Madanha in ^[17] showed that if a non-solvable group $G$ with $| \mathrm{cv}(G)| = 8$, then $G\cong {\mathrm{PSL}}_2(5)$ or $\mathrm{PGL}_2(5)$.

Some scholars are of interest in the set $\mathrm{cd}(G)$. For example, the structure of finite groups are determined if the degrees of finite groups are either prime powers ^[13,19,20] or divisible by two prime divisors ^[21] or the direct product of at most two primes ^[9,16] or square-free ^[12] or $p'$-numbers ^[6] or are consecutive ^[8,14,15,22].

In this paper, we go on the subgroup's character values and group structure, namely, we replace the condition "the number of character values of a group is small" with the condition "the number of character values of each proper subgroup of a group is small". For convenient arguments, we introduce the following definition.

Definition 1.1. Let $\sum G$ be the set of the proper subgroups of a group $G$, and $n$ a positive integer. A group is called a ${\bf{pcv}}_n$-group if for each $H\in\sum G$, $| \mathrm{cv}(H)|\leq n$.

We mainly show the following.

Theorem 1.2. If $G$ is a ${\bf{pcv}}_n$-group with $n\leq 5$, then $G$ is solvable.

Theorem 1.2 is corresponding to [10,Theorem 12.15] or [17,Theorem 1.1] which says that a finite group with $| \mathrm{cv}(G)|\leq 7$ or $| \mathrm{cd}(G)|\leq 3$ is solvable. Here we use the structure of a minimal simple group to prove Theorem 1.2.

We also obtain the following result which is corresponding to [17,Theorem 1.2] or [18,Theorem 2.2].

Theorem 1.3. Let $G$ be a non-solvable ${\bf{pcv}}_n$-group.

(1) If $n = 6$, then $G$ is isomorphic to $A_5$.

(2) If $n = 7$, then $G$ is isomorphic to ${\mathrm{PSL}}_2(q)$ with $q\in\{5, 7\}$.

The structure of this short paper is as follows. In Section 2, some basic results are given, and in Section 3, the structures of non-solvable ${\bf{pcv}}_6$- and ${\bf{pcv}}_7$-groups are identified respectively. For the other notions and symbols are standard, please see ^[5,10].

2.   Minimal simple groups

In this section, we assemble some results needed. First result is due to Madanha.

Lemma 2.1. [17,Theorem 1.1] If $| \mathrm{cv}(G)|\leq 7$, then $G$ is solvable.

A minimal simple group is a simple group of composite order all of whose proper subgroups are solvable.

Lemma 2.2. [25,Corollary 1] Every minimal simple group is isomorphic to one of the following minimal simple groups:

(1) ${\mathrm{PSL}}_2(2^p)$ for $p$ a prime;

(2) ${\mathrm{PSL}}_2(3^p)$ for $p$ an odd prime;

(3) ${\mathrm{PSL}}_2(p)$, for $p$ any prime exceeding 3 such that $p^2+1\equiv0\pmod5$;

(4) $Sz(2^p)$ for $p$ an odd prime;

(5) ${\mathrm{PSL}}_3(3)$.

Let $A$ be a group, and let $\exp A$ be a number which is minimal such that the order of all elements from $A$ divides $\exp A$. The following two lemmas are given because the subgroups of a group can control the structure of a group.

Lemma 2.3. Let $G$ be a dihedral group $D_{2n}$ of order $2n$. Then $| \mathrm{cv}(G)|\leq n+1$.

Proof. Obviously $\exp A\leq n$ with equality when $A$ is cyclic and so by [17,Lemma 2.2], $| \mathrm{cv}(A)|\leq n$, and so $| \mathrm{cv}(G)|\leq n+1$.

Remark 2.4. In Lemma 2.3 we cannot replace $\leq$ with $=$. For instance, Let $n = 16$, then by ^[4], $\mathrm{cv}(D_{32}) = \{1, 2, -1, -2, 0, A, -A, B, -B, C, -C\}$. Now $| \mathrm{cv}(D_{32})| = 11 < 16$.

Remark 2.5. The result of Lemma 2.3 is mostly possible. For example, let $n = 5$. Then by ^[4], $\mathrm{cv}(D_{10}) = \{1, 0, -1, 2, A, A^*\}$. Now $| \mathrm{cv}(D_{10})| = 6 = 5+1$.

Lemma 2.6. Let $G$ be a Frobenius group with the form $E_{p^n}:C_{p^n-1}$ where $n\geq1$ is a positive integer. Then $| \mathrm{cv}(G)| = p+1$. In particular, if $E_{p^n}:C_k$ is a Frobenius subgroup of $E_{p^n}:C_{p^n-1}$, then $| \mathrm{cv}(E_{p^n}:C_k)|\leq p+1$.

Proof. We see $\exp E_{p^n} = p$, so Lemma 2.2 of ^[17] forces $\mathrm{cv}(E_{p^n}) = p$. We know that $C_{p^n-1}$, acts fixed-point-freely on $E_{p^n}$ so by Theorem 18.7 of ^[8], $| \mathrm{cv}(G)| = p+1$.

Lemma 2.7. (1) $G$ is a ${\bf{pcv}}_n$-group $G$ if and only if for $H\in \sum G$, $| \mathrm{cv}(H)|\leq n$.

(2) Let $N$ be a normal subgroup of a ${\bf{pcv}}_n$-group $G$, then, both $N$ and $G/N$ are ${\bf{pcv}}_n$-groups.

Proof. We conclude the two results from the definition of a ${\bf{pcv}}_n$-group.

3.   Simple ${\bf{pcv}}_n$-groups and Solvable ${\bf{pcv}}_5$-groups

In this section we will first determine the structure of simple ${\bf{pcv}}_n$-groups for $n = 6, 7$ by using Lemma 2.2 and then show the solvability of ${\bf{pcv}}_n$-groups when $n\leq 5$. For easy reading, we rewrite Theorem 1.3 here.

Theorem 3.1. Let $G$ be a non-abelian simple ${\bf{pcv}}_n$-group with $n\leq 7$. Then

(1) if $n = 6$, $G$ is isomorphic to $A_5$;

(2) if $n = 7$, $G$ is isomorphic to $A_5$ or ${\mathrm{PSL}}_2(7)$.

Proof. We know that for each $H\in\sum G$, $| \mathrm{cv}(H)|\leq 7$, $H$ is solvable by Lemma 2.1. It follows that $G$ is a group whose proper subgroups are all solvable, so we can assume that $G$ is a minimal simple group. Thus $G$ is isomorphic to ${\mathrm{PSL}}_2(2^p)$ for $p$ a prime; ${\mathrm{PSL}}_2(3^p)$ for $p$ an odd prime; ${\mathrm{PSL}}_2(p)$, for $p$ any prime exceeding 3 such that $p^2+1\equiv0\pmod5$; $Sz(2^p)$ for $p$ an odd prime; ${\mathrm{PSL}}_3(3)$; see Lemma 2.2. So in what follows, these cases are considered.

Case 1: ${\mathrm{PSL}}_2(q)$ for certain $q$.

By Table 1, $D_{2(q+1)/k}\in\max {\mathrm{PSL}}_2(q)$ and so by Lemma 2.3, $\frac{q+1}{2}\leq 7$ when $q$ is odd or $q+1\leq 7$ when $q$ is even. It follows that $q$ is equal to 4, 5, 7, 9, 11 or 13. Note that ${\mathrm{PSL}}_2(4)\cong {\mathrm{PSL}}_2(5) \cong A_5$, so two subcases are dealt with.

Subcase 1: $q\in\{5, 7\}$.

Let $q = 5$. Then by [5,p. 2], $\max A_5 = \{A_4, D_{10}, S_3\}$ and by ^[4], $\mathrm{cv}(A_4) = \{-1, 0, 1, A, A^*\}$, $\mathrm{cv}(D_{10}) = \{-1, 0, 1, 2, A, A^*\}$ and $\mathrm{cv}(S_3) = \{-1, 0, 1, 2\}$. It follows that for each $H\in\max A_5$, $| \mathrm{cv}(H)|\leq 6$, so $G$ is isomorphic to $A_5$.

If $q = 7$, then $\max {\mathrm{PSL}}_2(7) = \{S_4, 7:3\}$ and by ^[4], $\mathrm{cv}(7:3) = \{1, 3, A, A^*, B, B^*, 0\}$, $\mathrm{cv}(S_4) = \{1, 2, 3, -1, 0\}$, so $| \mathrm{cv}(7:3)| = 7$ and $| \mathrm{cv}(S_4)| = 5$. Assumption shows that $G$ is isomorphic to ${\mathrm{PSL}}_2(7)$ as desired.

Subcase 2: $q\in\{9, 11, 13\}$.

By Table 1, $A_5\in\max {\mathrm{PSL}}_2(q)$ when $q = 9$ or $11$ and $| \mathrm{cv}(A_5)| = 8$ by [17,Theorem 1.2] or [5,p. 2].

If $q = 13$, then $13:6\in\max {\mathrm{PSL}}_2(13)$ and $\mathrm{cv}(13:6) = \{$1, 6, $A$, $A^*$, $B$, $-B$, $B^*$, $-B^*$, -1, 0$\}$ by ^[4], so $| \mathrm{cv}(13:6)| = 10$, a contradiction.

It follows that ${\mathrm{PSL}}_2(q)$ for $q\in\{9, 11, 13\}$ is not a ${\bf{pcv}}_n$-group with $n\leq7$.

Case 2: $Sz(2^p)$ for $p$ an odd prime.

In this case, by Table 2, $D_{2(q-1)}\in\max Sz(q)$, so by Lemma 2.3, $q-1\leq 7$, so $q = 8$. Now $2^{3+3}:7\in\max Sz(8)$ and by ^[4], $\mathrm{cv}(2^{3+3}:7) = \{$1, 7, 14, $-2$, $-1$, $A$, $-A$, $B$, $B^*$, $C$, $C^*$, $D$, $D^*$, 0$\}$. Thus $| \mathrm{cv}(2^{3+3}:7)| = 14$, a contradiction.

Case 3: ${\mathrm{PSL}}_3(3)$.

By [5,pp. 13], $13:3\in{\mathrm{PSL}}_3(3)$ and so by ^[4], $\mathrm{cv}(13:3) = \{0, 1, 3, A, A^*, B, B^*, C, C^*\}$. Now $| \mathrm{cv}(13:3)| = 9\nleq7$, a contradiction.

Now we can prove Theorem 1.2. For reader's convenience, we rewrite it here.

Theorem 3.2. If $G$ is a ${\bf{pcv}}_n$-group with $n\leq 5$, then $G$ is solvable.

Proof. By hypothesis, we know that every proper subgroup of $G$ is solvable. If $G$ is non-solvable, then we can assume that $G$ is simple. Now as the proof of Theorem 3.1 we obtain a contradiction. Thus $G$ is solvable.

4.   Non-solvable ${\bf{pcv}}_n$-groups

In this section, we first show the structure of non-solvable ${\bf{pcv}}_6$-groups and then the structures of non-solvable ${\bf{pcv}}_7$-groups are determined. For convenient reading, we rewrite Theorem 1.3 here.

Theorem 4.1. Let $G$ be a non-solvable ${\bf{pcv}}_n$-group.

(1) If $n = 6$, then $G$ is isomorphic to $A_5$

(2) If $n = 7$, then $G$ is isomorphic to ${\mathrm{PSL}}_2(q)$ with $q\in\{5, 7\}$.

Proof. The non-solvability of $G$ shows that $G$ has a normal sequel $1\leq H\leq K\leq G$ such that $K/H$ is isomorphic to a direct product of isomorphic simple groups and that $|G/K|$ divides $|\mathrm{Out}(K/H)|$, where $\mathrm{Out}(A)$ denotes the outer-automorphism group of a group $A$; see ^[27].

Now we have that

where $S\cong A_5$ when $n = 6$ and $S\cong A_5$ or ${\mathrm{PSL}}_2(7)$ when $n = 7$; see Theorem 3.1. By Lemma 2.7, $K/H$ is a ${\bf{pcv}}_n$-group, so let $H\in\max S$, now

If $m\geq2$, then $| \mathrm{cv}(A_5)| = 8$ and $| \mathrm{cv}({\mathrm{PSL}}_2(7))| = 10$ as $\mathrm{cv}(A_5) = \{1, 3, 4, 5, -1, 0, A, A^*\}$ and $\mathrm{cv}({\mathrm{PSL}}_2(7)) = \{1, 3, 6, 7, 8, -1, 2, 0, A, A^*\}$ by [5,p. 2-3], so $| \mathrm{cv}(H\times S\times\cdots\times S)|\geq 8$, a contradiction. Thus $m = 1$ and for any $S < H\leq \mathrm{Aut}(S)$, $| \mathrm{cv}(S)|\geq8$ shows that $H$ is not a ${\bf{pcv}}_n$-group with $n = 6, 7$; see [17,Theorem 1.2]. Now $G/H\cong A_5$ or ${\mathrm{PSL}}_2(7)$ and $G$ is not an almost simple group. It follows that

see [7,Chap 2,Theorem 6.10].

So in the following two cases are done with.

Where $\mathrm{N}1$ maximal under $< \delta >$ with $|\delta| = (q-1, 2)$; $\mathrm{N}2$ maximal under subgroups not contained in $< \varphi >$ with $|\varphi| = e$, $q = p^e$, $p$ a prime.

Case 1: $G'/H\cong A_5$ or $\mathrm{SL}_2(5)$.

By Table 3, $2.A_4\in\max \mathrm{SL}_2(5)$ and by ^[4], $\mathrm{cv}(2.A_4) = \{1, 3, 2, -2, 0, A, A^*, -A^*, -A\}$. It follows that $| \mathrm{cv}(2.A_4)| = 9 > 7$, so $\mathrm{SL}_2(5)$ is not a ${\bf{pcv}}_n$-group with $n = 6, 7$. Thus $G'/H\cong \mathrm{SL}_2(5)$ is impossible. Now $G'/H\cong A_5$ and $G'\cap H = 1$, so $[G', H]\leq G'\cap H = 1$. Thus,

It follows that $G\cong H\times A_5$. If $H\neq 1$, then $A_5\in\sum G$. Observe that $| \mathrm{cv}(A_5)| = 8$, so in this case $G$ is a non-${\bf{pcv}}_n$-group with $n = 6, 7$. Therefore $G$ is isomorphic to $A_5$, the desired result.

Case 2: $G'/H\cong {\mathrm{PSL}}_2(7)$ or $\mathrm{SL}_2(7)$.

In this case $n = 7$. Table 3 gives that $E_7:C_6\in \max \mathrm{SL}_2(7)$ and by ^[4], $\mathrm{cv}(E_7:C_6) = \{1, 6, -1, 0, A, -A, A^*, -A^*\}$. It follows that $| \mathrm{cv}(E_7:C_6)| = 8$ (this result can be gotten from Lemma 2.6), so $\mathrm{SL}_2(7)$ is not a ${\bf{pcv}}_7$-group. Thus $G'/H\cong \mathrm{SL}_2(7)$ is not possible. Now consider when $G'/H\cong {\mathrm{PSL}}_2(7)$. Then $G'\cap H = 1$, $[G', H] = 1$ and $G\cong H\times {\mathrm{PSL}}_2(7)$ too. If $H\neq1$, then ${\mathrm{PSL}}_2(7)$ is not a ${\bf{pcv}}_7$-group, so $H = 1$ and $G$ is isomorphic to ${\mathrm{PSL}}_2(7)$, the wanted result.

Proposition 4.2. Let $G$ be a ${\bf{pcv}}_n$-group with $n\leq 7$. Assume that $G$ has no section isomorphic to ${\mathrm{PSL}}_2(q)$ for $q\in\{5, 7\}$, then $G$ is solvable.

Proof. By Theorems 1.2 and 1.3, we can get the desired result.

Acknowledgments

The authors were supported by NSF of China(Grant No: 11871360) and also the first author was supported by the Opening Project of Sichuan Province University Key Laborstory of Bridge Non-destruction Detecting and Engineering Computing (Grant Nos: 2022QYJ04), and by the the Project of High-Level Talent of Sichuan Institute of Arts and Science (Grant No: 2021RC001Z).

Conflict of interest

The authors declare that they have no conflicts of interest.