A new characterization of the automorphism groups of Mathieu groups

Xin Liu; Guiyun Chen; Yanxiong Yan

doi:10.1515/math-2021-0112

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A new characterization of the automorphism groups of Mathieu groups

Xin Liu , Guiyun Chen and Yanxiong Yan

Published/Copyright: December 9, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

Let cd ( G ) be the set of irreducible complex character degrees of a finite group G . ρ ( G ) denotes the set of primes dividing degrees in cd ( G ) . For any prime p, let p e p ( G ) = max { χ ( 1 ) p ∣ χ ∈ Irr ( G ) } and V ( G ) = { p e p ( G ) ∣ p ∈ ρ ( G ) } . The degree prime-power graph Γ ( G ) of G is a graph whose vertices set is V ( G ) , and two vertices x , y ∈ V ( G ) are joined by an edge if and only if there exists m ∈ cd ( G ) such that x y ∣ m . It is an interesting and difficult problem to determine the structure of a finite group by using its degree prime-power graphs. Qin proved that all Mathieu groups can be uniquely determined by their orders and degree prime-power graphs. In this article, we continue this topic and successfully characterize all the automorphism groups of Mathieu groups by using their orders and degree prime-power graphs.

Keywords: finite group; character degree; degree prime-power graph; automorphism group

MSC 2010: 20C33

1 Introduction

The groups involved in this paper are all finite groups, and all characters are complex characters.

We use Irr ( G ) to denote the set of all complex irreducible characters of a group G , and cd ( G ) = { χ ( 1 ) ∣ χ ∈ Irr ( G ) } to denote the set of all irreducible character degrees of G forgetting multiplicities. In particular, cd ∗ ( G ) denotes a multiple set whose elements can be repeated and ∣ cd ∗ ( G ) ∣ = ∣ Irr ( G ) ∣ . Denote by ρ ( G ) the set of primes dividing degrees in cd ( G ) . We use Out ( G ) to denote the outer automorphism group of a group G . We use π ( n ) to denote the set of all prime divisors of n and n p the maximum power of p such that n p ∣ n , where n is a positive integer and p is a prime. Let P ( n ) = { n p ∣ p ∈ π ( n ) } . For convenience, we write P ( G ) = P ( ∣ G ∣ ) and π ( G ) = π ( ∣ G ∣ ) . All other symbols and terms are standard (see [1,2]).

In 2000, Huppert proposed the following conjecture:

Huppert’s conjecture: Let M be a non-abelian simple group such that cd ( G ) = cd ( M ) , then G ≅ M × A , where A is an abelian group.

Huppert conjectured that all finite non-abelian simple groups can be uniquely determined by their orders and the sets of irreducible character degrees. Huppert checked the conjecture case-by-case for many non-abelian simple groups such as Sz ( q ) , the alternating groups A n with 5 ≤ n ≤ 11 , and most of the sporadic simple groups and some other simple groups of Lie type (see [3,4,5]). Tong-Viet and Wakefield proved that if M is one of G 2 ( q ) ( q ≥ 7 ) , PSL 3 ( q ) , PSU 3 ( q 2 ) , G 2 2 ( q 2 ) , and PSp 4 ( q ) ( q > 7 ) , then the conjecture holds (see [6,7, 8,9]). Nguyen continued Huppert’s work and proved the conjecture for the alternating groups A n with n = 12 , 13 (see [10]). Bessenrodt et al. solved the remaining alternating groups and showed that Huppert’s conjecture follows for all the simple alternating groups A n ( n ≥ 14 ) (see [11]). In addition, Daneshkhah proved that all the sporadic simple groups Co 1 , Co 2 , Co 3 , and Fi 23 can be uniquely characterized by the set of their irreducible character degrees (see [12,13]). However, up to now, Huppert’s conjecture is still open. So, a natural problem is what the influence on the structure of a finite group is by weakening the condition of Huppert’s conjecture. In particular, an interesting question is that if G and M have the same order, when we just consider some subset of cd ( M ) , whether we can determine the structure of such finite groups? Some people found that many non-abelian simple groups can be characterized by their orders and some largest irreducible character degrees. For example, Xu proved that simple K 3 -groups and Mathieu groups can be uniquely determined by their orders and one or two irreducible character degrees (see [14,15]). In addition, Heydari characterized simple K 4 -groups according to their orders and at most three distinct irreducible character degrees (see [16]).

The character degree graph of G , which is denoted by Δ ( G ) , is a graph with the vertex set ρ ( G ) and two vertices x and y are adjacent in Δ ( G ) , if there is some f ∈ cd ( G ) such that x y ∣ f (see [17]). Many researchers began to investigate the relationship between Δ ( G ) and structure of finite group, trying to know about the properties of Δ ( G ) . Khosravi et al. proved that A 5 , A 6 , A 7 , A 8 , L 3 ( 3 ) , L 3 ( 4 ) , L 2 ( 64 ) , L 2 ( q ) (where q is an odd prime or a square of an odd prime, and q ⩾ 5 ), and L 2 ( 2 α ) (where α is a positive integer such that 2 α − 1 or 2 α + 1 is a prime) can be determined by their orders and character degree graphs (see [18,19]). Furthermore, the authors also proved that some simple groups of orders less than 6,000 are uniquely determined by their orders and character degree graphs (see [20]).

But an interesting fact is that not all non-abelian simple groups can be uniquely determined by their orders and character degree graphs. We knew that the Mathieu M 11 , M 22 , and M 23 can be uniquely determined by their orders and character degree graphs, while M 12 cannot be determined by the order of M 12 and the character degree graph Δ ( M 12 ) (see [21]). In fact, M 12 and A 4 × M 11 have the same orders and the same character degree graphs. So it is a difficult problem whether there exists some graph such that any Mathieu group can be uniquely determined by the graph. Based on this fact, in 2018, Qin et al. for the first time put forward the degree prime-power graph via the set of irreducible character degrees. Also, the authors successfully characterized all the Mathieu groups and sporadic simple groups just by using their orders and character degree prime-power graphs (see [22,23]). In this article, we continue this topic and prove that all the automorphism groups of Mathieu groups can also be uniquely determined by their orders and character degree prime-power graphs.

We first give the following definition:

Definition 1.1

Let G be a group. For every p ∈ ρ ( G ) , let p e p ( G ) = max { χ ( 1 ) p ∣ χ ∈ Irr ( G ) } and V ( G ) = { p e p ( G ) ∣ p ∈ ρ ( G ) } . Define the degree prime-power graph Γ ( G ) as follows: V ( G ) is the vertex set, and there is an edge between distinct numbers x , y ∈ V ( G ) if x y divides some integer in cd ( G ) . Denote the edge between distinct numbers x , y ∈ V ( G ) by x ∼ y , and the set of all edges of Γ ( G ) by E ( G ) .

Definition 1.2

Let G be a group, for any p ∈ ρ ( G ) . We define v p ( G ) the p -exponent variation of G as follows: v p ( G ) = min log p ∣ G ∣ χ ( 1 ) p ∣ χ ( 1 ) ∈ Irr ( G ) . Obviously, v p ( G ) = log p ( ∣ G ∣ p ) − e p ( G ) .

Definition 1.3

Let G be a group and p ∈ π ( G ) . An irreducible character χ has p -defect zero if χ ( 1 ) p = ∣ G ∣ p .

In this article, we successfully characterize all the automorphism groups of Mathieu groups by using their orders and degree prime-power graphs. Our main result is:

Theorem 1.4

Let G be a group. Suppose that M is one of the automorphism groups of a Mathieu group. Then G ≅ M if and only if ∣ G ∣ = ∣ M ∣ and Γ ( G ) = Γ ( M ) .

2 Preliminary results

In what follows, we need to make some preparations for the proof of Theorem 1.4 and we begin with some important lemmas which will be used in what follows.

Lemma 2.1

[22] Suppose there is an irreducible character of G with p -defect zero for some p ∈ π ( G ) . If N ⊴ ⊴ G and p ∈ π ( N ) , then there is an irreducible character of N which has p -defect zero, and O p ( N ) = 1 .

Lemma 2.2

[22] Let G be a finite solvable group of order p 1 α 1 p 2 α 2 ⋯ p n α n , where p 1 , p 2 , … , p n are distinct primes. If k p n + 1 ∤ p i α i for each 1 ⩽ i ⩽ n − 1 and k ∈ Z ∗ , then the Sylow p n -subgroup is normal in G .

Lemma 2.3

[22] Let G be a non-solvable group. If T / S is a non-abelian chief factor of G , then there is a normal series 1 ⩽ H < K ⩽ G such that K / H ≅ T / S and ∣ G / K ∣ ∣ ∣ Out ( T / S ) ∣ .

Lemma 2.4

[23] Let G be a group. For any p ∈ π ( G ) \ ( 2 ) , we have v p ( G ) = 0 , but v 2 ( G ) = 2 . If N is an arbitrary non-unit and solvable subnormal subgroup of group G , then N satisfies one of the following conditions:

∣ N ∣ = 2 or 4;
∣ N ∣ = 2 3 , and cd ∗ ( N ) = { 1 , 1 , 1 , 1 , 2 } ;
∣ N ∣ = 2 2 ⋅ 3 , and cd ∗ ( N ) = { 1 , 1 , 1 , 3 } ;
∣ N ∣ = 2 3 ⋅ 3 , and cd ∗ ( N ) = { 1 , 1 , 1 , 2 , 2 , 2 , 3 } or cd ∗ ( N ) = { 1 , 1 , 2 , 3 , 3 } ;
∣ N ∣ = 2 4 ⋅ 3 , and cd ∗ ( N ) = { 1 , 1 , 2 , 2 , 2 , 2 , 3 , 3 , 4 } .

Lemma 2.5

[22] Let G be a group of order 672 = 2 5 ⋅ 3 ⋅ 7 . If 3 , 7 ∈ V ( G ) , then one of the following holds:

The number of irreducible characters with same degree is less than 11;
G has a normal abelian Sylow 2-subgroup.

Corollary 2.6

[22] Let G be a non-solvable group. Then there is a subnormal series

G = G 0 ⩾ G 1 > G 2 ⩾ ⋯ ⩾ G 2 k − 1 > G 2 k ⩾ 1 ( k ⩾ 1 )

such that G 2 k is solvable, G 2 i − 1 / G 2 i is a non-abelian chief factor of G 2 i − 2 and ∣ G 2 i − 2 / G 2 i − 1 ∣ ∣ ∣ Out ( G 2 i − 1 / G 2 i ) ∣ for each 1 ⩽ i ⩽ k .

3 Proof of Theorem 1.4

Remark

If G is a simple group, then G ≅ Inn ( G ) . By [1], it is easy to check that ∣ Out ( M 11 ) ∣ = 1 , ∣ Out ( M 23 ) ∣ = 1 and ∣ Out ( M 24 ) ∣ = 1 . Hence, in what follows we just need to discuss the automorphism groups of the Mathieu groups M 12 and M 22 .

Proof of Theorem 1.4

The necessity of Theorem 1.4 is obvious and so it is enough to prove the sufficiency. In what follows, we write up the proof of what M is case-by-case.

Case 1. M = Aut ( M 12 ) .

From [1], we have ∣ G ∣ = 2 7 ⋅ 3 3 ⋅ 5 ⋅ 11 , V ( G ) = { 2 5 , 3 3 , 5 , 11 } , and E ( G ) = { 5 ∼ 11 } . By Lemma 2.1, O 3 ( G ) = O 5 ( G ) = O 11 ( G ) = 1 . We claim that G is non-solvable. Otherwise, if the group G is solvable, a Sylow 11-subgroup G 11 of G is normal in G by Lemma 2.2. This is in contradiction with O 11 ( G ) = 1 .

By Lemma 2.3, there is a normal series 1 ⩽ H < K ⩽ G such that K / H is a non-abelian chief factor of G and ∣ G / K ∣ ∣ ∣ Out ( K / H ) ∣ . By comparing the order of G and the orders of the simple groups in [1], every non-abelian chief factor of G is isomorphic to one of the following groups: A 5 , A 6 , L 2 ( 11 ) , M 11 , and M 12 . And the non-abelian chief factors of G are pairwise non-isomorphic.

If K / H ≅ A 5 , by ∣ A 5 ∣ = 2 2 ⋅ 3 ⋅ 5 , and ∣ Out ( A 5 ) ∣ = 2 , then ∣ G / K ∣ ∣ 2 , which implies that ∣ H ∣ = 2 α ⋅ 3 2 ⋅ 11 , where α = 4 or 5. Since the order of the group H cannot be divisible by the order of any non-abelian simple group, H is solvable. By Lemma 2.2 a Sylow 11-subgroup of H is normal in H , and this is in contradiction with O 11 ( G ) = 1 .

If K / H ≅ A 6 , by ∣ A 6 ∣ = 2 3 ⋅ 3 2 ⋅ 5 , and ∣ Out ( A 6 ) ∣ = 4 , we have ∣ G / K ∣ ∣ 4 , and so ∣ H ∣ = 2 α ⋅ 3 ⋅ 11 , where α = 2 , 3, or 4. For the same reasons as above, we also get a contradiction.

If K / H ≅ L 2 ( 11 ) , since ∣ L 2 ( 11 ) ∣ = 2 2 ⋅ 3 ⋅ 5 ⋅ 11 and ∣ Out ( L 2 ( 11 ) ) ∣ = 2 , then ∣ G / K ∣ ∣ 2 . One has that ∣ H ∣ = 2 α ⋅ 3 2 , where α = 4 or 5. Then H is solvable. By Lemma 2.4, it is easy to deduce that there exists no such group ∣ H ∣ such that H satisfies the above condition, and hence H is unsolvable, which leads to a contradiction.

Assume that K / H ≅ M 11 , by ∣ M 11 ∣ = 2 4 ⋅ 3 2 ⋅ 5 ⋅ 11 , and ∣ Out ( M 11 ) ∣ = 1 , ∣ G / K ∣ ∣ 1 , we have ∣ K ∣ = ∣ G ∣ , G / H ≅ M 11 , and ∣ H ∣ = 2 3 ⋅ 3 . By 3 3 ∈ V ( G ) and Lemma 2.1, there is a character θ ∈ Irr ( H ) such that 3 ∣ θ ( 1 ) , then θ ( 1 ) = 3 . According to v 3 ( H ) = v 3 ( G ) = 0 , then θ is G -invariant. Since the Schur multiplier H ( M 11 , C × ) of M 11 is 1, θ is extendible to G by [2, Theorem 11.7]. Note that 3 2 ⋅ 5 ∈ cd ( M 11 ) . Then, we have 3 3 ⋅ 5 ∈ cd ( G ) by [2, Corollary 6.17]. This is in contradiction with 3 3 ∼ 5 ∉ E ( G ) .

Now, assume that K / H ≅ M 12 , by ∣ M 12 ∣ = 2 6 ⋅ 3 3 ⋅ 5 ⋅ 11 , and ∣ Out ( M 12 ) ∣ = 2 , ∣ G / K ∣ ∣ 2 . This means that ∣ H ∣ = 2 α , where α = 0 or 1. By N / C theorem, we have G / H ≲ Aut ( K / H ) . So we get K / H ≲ G / H ≲ Aut ( K / H ) , that is, M 12 ≲ G / H ≲ Aut ( M 12 ) .

If G / H ≅ Aut ( M 12 ) , and since ∣ G ∣ = ∣ Aut ( M 12 ) ∣ , we deduce H = 1 and G ≅ Aut ( M 12 ) .

If G / H ≅ M 12 , then ∣ H ∣ = 2 . So H ≤ Z ( G ) . Therefore, G is a central extension of Z 2 by M 12 and G is isomorphic to one of the following groups:

2 ⋅ M 12 ( a non-split extension of Z 2 by M 12 ) , Z 2 × M 12 ( a split extension of Z 2 by M 12 ) .

If G ≅ 2 ⋅ M 12 , by [1], we have χ ∈ Irr ( 2 ⋅ M 12 ) such that χ ( 1 ) ∣ 2 = 2 6 . This is in contradiction with V ( G ) .

If G ≅ Z 2 × M 12 , by [1] and [2, Theorem 4.21], we have χ ∈ Irr ( Z 2 × M 12 ) such that χ ( 1 ) ∣ 2 = 2 4 . This is in contradiction with V ( G ) .

Case 2. M = Aut ( M 22 ) .

From [1], we have ∣ G ∣ = 2 8 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 , V ( G ) = { 2 4 , 3 2 , 5 , 7 , 11 } , and E ( G ) = { 2 4 ∼ 5 , 2 4 ∼ 7 , 3 2 ∼ 5 , 3 2 ∼ 11 , 5 ∼ 7 , 5 ∼ 11 , 7 ∼ 11 } . By Lemma 2.1, O 3 ( G ) = O 5 ( G ) = O 7 ( G ) = O 11 ( G ) = 1 . We claim that G is non-solvable. Otherwise, if the group G is solvable, a Sylow 11-subgroup G 11 of G is normal in G by Lemma 2.2. This is in contradiction with O 11 ( G ) = 1 .

By Lemma 2.3, there is a normal series 1 ⩽ H < K ⩽ G such that K / H is a non-abelian chief factor of G and ∣ G / K ∣ ∣ ∣ Out ( K / H ) ∣ . Since ∣ G ∣ = 2 8 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 , the non-abelian chief factors of G are pairwise non-isomorphic, and by [1], we see that K / H is isomorphic to one of the following groups: A 5 , L 3 ( 2 ) , A 6 , L 2 ( 8 ) , L 2 ( 11 ) , A 7 , M 11 , A 8 , L 3 ( 4 ) , and M 22 .

The orders of the outer automorphisms of these simple groups are not divisible by 11. We claim that there is a non-abelian simple chief factor of G whose order is divisible by 11. Otherwise, there is a solvable subnormal subgroup N of G such that 11 ∣ ∣ N ∣ by Corollary 2.6. Then N has a normal Sylow 11-subgroup by Lemma 2.2. This contradicts O 11 ( N ) = 1 by Lemma 2.1. Hence, G has a chief factor isomorphic to L 2 ( 11 ) , M 11 , or M 22 .

Assume that G has a chief factor isomorphic to L 2 ( 11 ) . By Lemma 2.3, there is a chief factor K / H of G such that K / H ≅ L 2 ( 11 ) . According to ∣ L 2 ( 11 ) ∣ = 2 2 ⋅ 3 ⋅ 5 ⋅ 11 , and ∣ Out ( L 2 ( 11 ) ) ∣ = 2 , ∣ G / K ∣ ∣ 2 , this shows that ∣ H ∣ = 2 α ⋅ 3 ⋅ 7 , where α = 5 or 6. Since the edge 5 ∼ 11 ∈ E ( G ) , we have that 5 ∼ 11 ∈ E ( K ) by [2, Corollary 11.29]. Hence, there exists a character χ ∈ K such that 55 ∣ χ ( 1 ) . Let ξ ∈ Irr ( H ) be an irreducible constituent of χ H . By [2, Theorem 6.2], we see that χ H = e ∑ i = 1 s ξ i where ξ = ξ 1 , ξ 2 , … , ξ s are the distinct conjugates of ξ in K and e = [ χ H , ξ ] . If s = ∣ K : I K ( ξ ) ∣ ≠ 1 , then there is a maximal subgroup T / H of K / H such that I K ( ξ ) / H ≤ T / H . Since the index of the maximal subgroup of L 2 ( 11 ) is 11, 12, or 55 (see [1]), s is divisible by 11, 12, or 55. Since 3 2 , 7 ∈ V ( G ) , we know 3 , 7 ∈ V ( H ) .

If α = 5 , ∣ H ∣ = 2 5 ⋅ 3 ⋅ 7 . By Lemma 2.5, we have s < 11 , then s = 1 , χ H = e ξ . Since ξ ( 1 ) ∣ ∣ H ∣ , 5 ⋅ 11 ∤ ξ ( 1 ) , then 55 ∣ e and 5 5 2 ≤ [ χ H , χ H ] ≤ ∣ K : H ∣ [ χ , χ ] = 2 2 ⋅ 3 ⋅ 5 ⋅ 11 , where χ H is the restriction of χ to H , a contradiction. Therefore, H has a normal abelian Sylow 2-subgroup H 2 . By [2, Theorem 6.15], we have 2 4 ∣ ∣ G : H 2 ∣ , i.e., 2 4 ∣ 2 3 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 , a contradiction too.

If α = 6 , then ∣ H ∣ = 2 6 ⋅ 3 ⋅ 7 . Noting that ∣ G ∣ = ∣ K ∣ and s = ∣ G : I G ( ξ ) ∣ , we have χ ( 1 ) = e s ξ ( 1 ) . Since ξ ( 1 ) ∣ ∣ H ∣ , it follows that 5 ⋅ 11 ∤ ξ ( 1 ) , and hence 5 ⋅ 11 ∣ e s . If H is solvable, there exists N ⊴ H such that the normal subgroup of H / N is elementary abelian, so that ∣ N ∣ 2 ⩽ 2 6 and ∣ ( H / N ) ∣ 2 ⩽ 2 6 . Since ∣ GL ( 6 , 2 ) ∣ is indivisible by 11, it implies that L 2 ( 11 ) induces the identity on N and H / N , and so L 2 ( 11 ) acts trivially on H . Consequently,

G = ( H ∘ SL 2 ( 11 ) ) . O or ( H × L 2 ( 11 ) ) . O with O ⩽ 2 .

It follows that H ⩽ I G ( ξ ) , and hence χ can be viewed as an irreducible character of G / H , which is impossible because 55 divides χ ( 1 ) .

Thus, H is non-solvable. By Lemma 2.3 and [1], there is a normal series 1 ⩽ S < T ⩽ H such that T / S is a non-abelian chief factor of H , and T / S ≅ L 2 ( 7 ) , we get ∣ S ∣ = 2 t , where t = 3 or 2. If t = 3 , then H / S ≅ L 2 ( 7 ) , and hence H ≅ AGL 3 ( 2 ) or S × L 2 ( 7 ) . According to C G ( H ) ⊆ I G ( ξ ) , then s ∣ ∣ G : C G ( H ) ∣ = ∣ G : I G ( ξ ) ∣ ⋅ ∣ I G ( ξ ) : C G ( H ) ∣ . By N / C theorem, we have G / C G ( H ) ≲ Aut ( H ) , that is, s ∣ ∣ Aut ( H ) ∣ . In either case, 55 ∤ ∣ Aut ( H ) ∣ , then 55 ∣ e , again a contradiction. In a similar fashion, we rule out the possibility where t = 2 .

Assume that G has a chief factor isomorphic to M 11 . By ∣ M 11 ∣ = 2 4 ⋅ 3 2 ⋅ 5 ⋅ 11 , ∣ Out ( M 11 ) ∣ = 1 , we have ∣ G / K ∣ ∣ 1 , ∣ K ∣ = ∣ G ∣ , then ∣ H ∣ = 2 4 ⋅ 7 . Since v 7 ( H ) = v 7 ( G ) = 0 , H has a G -invariant irreducible character η of degree 7. Since the Schur multiplier H ( M 11 , C × ) of M 11 is 1, η is extendible to G . Since 3 2 ∈ V ( M 11 ) , we see that 7 ∼ 3 2 ∈ E ( G ) by [2, Theorem 6.17]. This is in contradiction with 7 ∼ 3 2 ∉ E ( G ) .

Now, assume that K / H ≅ M 22 , by ∣ M 22 ∣ = 2 7 ⋅ 3 2 ⋅ 5 ⋅ 7 ⋅ 11 , ∣ Out ( M 22 ) ∣ = 2 , we know ∣ G / K ∣ ∣ 2 , then ∣ H ∣ = 2 α , where α = 0 or 1. By N / C theorem, we have G / H ≲ Aut ( K / H ) . So we get K / H ≲ G / H ≲ Aut ( K / H ) , that is M 22 ≲ G / H ≲ Aut ( M 22 ) .

If G / H ≅ Aut ( M 22 ) , and since ∣ G ∣ = ∣ Aut ( M 22 ) ∣ , we deduce H = 1 and G ≅ Aut ( M 22 ) .

If G / H ≅ M 22 , then ∣ H ∣ = 2 . So H ≤ Z ( G ) . Therefore, G is a central extension of Z 2 by M 22 and G is isomorphic to one of the following groups:

2 ⋅ M 22 ( a non-split extension of Z 2 by M 22 ) , Z 2 × M 22 ( a split extension of Z 2 by M 22 ) .

If G ≅ 2 ⋅ M 22 , by [1], we have χ ∈ Irr ( 2 ⋅ M 22 ) such that 3 2 ⋅ 7 ∣ χ ( 1 ) , then 3 2 ∼ 7 ∈ E ( 2 ⋅ M 22 ) . This is in contradiction with E ( G ) .

If G ≅ Z 2 × M 22 , by [1] and [2, Theorem 4.21], we have χ ∈ Irr ( Z 2 × M 22 ) such that χ ( 1 ) ∣ 2 = 2 3 , a contradiction to V ( G ) , which completes the proof of Theorem 1.4.□

Acknowledgements

The authors would like to express his deep gratitude to the referee for their or her invaluable comments and suggestions which helped to improve the paper.

Funding information: This work was partially supported by Natural Science Foundation of China (Grant Nos. 12071376 and 11971391), Natural Science Foundation Project of CQ CSTC (No. cstc2021jcyj-msxmX0426), and Fundamental Research Funds for the Central Universities (Nos. XDJK2019C116 and XDJK2019B030).
Conflict of interest: Authors state no conflict of interest.

References

[1] J. H. Conway , R. T. Curtis , S. P. Norton , R. A. Parker , and R. A. Wilson , Atlas of Finite Groups, Clarendon Press, New York, 1985. Search in Google Scholar

[2] I. M. Isaacs , Character Theory of Finite Groups, Academic Press, New York, 1976. Search in Google Scholar

[3] B. Huppert , Some simple groups which are determined by the set of their character degrees I, Illinois J. Math. 44 (2000), no. 4, 828–842. 10.1215/ijm/1255984694Search in Google Scholar

[4] B. Huppert , Some simple groups which are determined by the set of their character degrees II, Rend. Sem. Mat. Univ. Padova 115 (2006), 1–13. Search in Google Scholar

[5] B. Huppert , Some Simple Groups Which are Determined by The Set of Their Character Degrees. III–VIII, Institut für Experimentelle Mathematik, Universität Essen, 2000. 10.1215/ijm/1255984694Search in Google Scholar

[6] H. P. Tong-Viet and T. P. Wakefield , On Huppert’s conjecture for G2(q),q≥7 , J. Pure Appl. Algebra 216 (2012), no. 12, 2720–2729. 10.1016/j.jpaa.2012.03.028Search in Google Scholar

[7] T. P. Wakefield , Verifying Huppert’s conjecture for PSL3(q) and PSU3(q2) , Comm. Algebra 37 (2009), 2887–2906. 10.1080/00927870802625661Search in Google Scholar

[8] T. P. Wakefield , Verifying Huppert’s conjecture for G22(q2) , Algebr. Represent. Theory 14 (2011), 609–623. 10.1007/s10468-009-9206-xSearch in Google Scholar

[9] T. P. Wakefield , Verifying Huppert’s conjecture for PSp4(q) when q>7 , Algebr. Represent. Theory 15 (2012), 427–448. 10.1007/s10468-010-9246-2Search in Google Scholar

[10] H. N. Nguyen , H. P. Tong-Viet , and T. P. Wakefield , On Huppert’s conjecture for alternating groups of low degrees, Algebra Colloq. 22 (2015), no. 2, 293–308, https://doi.org/10.1142/S1005386715000267. Search in Google Scholar

[11] C. Bessenrodt , H. P. Tong-Viet , and J. P. Zhang , Huppert’s conjecture for alternating groups, J. Algebra 470 (2017), 353–378. 10.1016/j.jalgebra.2016.09.012Search in Google Scholar

[12] S. H. Alavi , A. Daneshkhah , H. P. Tong-Viet , and T. P. Wakefield , Huppert’s conjecture for Fi23 , Rend. Semin. Math. Univ. Padova 126 (2011), 201–211. 10.4171/RSMUP/126-11Search in Google Scholar

[13] S. H. Alavi , A. Daneshkhah , H. P. Tong-Viet , On Huppert’s conjecture for Conway and Fischer families of sporadic simple groups, J. Aust. Math. Soc. 94 (2013), no. 3, 289–303. 10.1017/S1446788712000535Search in Google Scholar

[14] H. J. Xu , G. Y. Chen , and Y. X. Yan , A new characterization of simple K3 -groups by their orders and large degrees of their irreducible characters, Comm. Algebra 42 (2014), no. 12, 5374–5380. 10.1080/00927872.2013.842242Search in Google Scholar

[15] H. J. Xu , Y. X. Yan , and G. Y. Chen , A new characterization of Mathieu-groups by the order and one irreducible character degree, J. Inequal. Appl. 2013 (2013), 209. 10.1186/1029-242X-2013-209Search in Google Scholar

[16] S. Heydari and N. Ahanjideh , Characterization of some simple K4 -groups by some irreducible complex character degrees, Int. J. Group Theory 5 (2016), no. 2, 61–74. Search in Google Scholar

[17] O. Manz , R. Staszewski , and W. Willems , On the number of components of a graph related to character degrees, Proc. Amer. Math. Soc. 103 (1988), no. 1, 31–37, https://doi.org/10.1090/S0002-9939-1988-0938639-1. Search in Google Scholar

[18] B. Khosravi , B. Khosravi , B. Khosravi , and Z. Momen , Recognition of some simple groups by character degree graph and order, Math. Rep. (Bucur.) 18 (2016), no. 68, 51–61. Search in Google Scholar

[19] B. Khosravi , B. Khosravi , B. Khosravi , and Z. Momen , Recognition of the simple group PSL(2,p2) by character degree graph and order, Monatsh. Math. 178 (2015), no. 2, 251–257. 10.1007/s00605-014-0678-3Search in Google Scholar

[20] B. Khosravi , B. Khosravi , B. Khosravi , and Z. Momen , Recognition by character degree graph and order of simple groups of order less than 6000, Miskolc Math. Notes 15 (2014), no. 2, 537–544.10.18514/MMN.2014.777Search in Google Scholar

[21] N. Ahanjideh and S. Heydari , Some simple groups which are determined by their character degree graphs, Sib. Elektron. Mat. Izv. 13 (2016), 1290–1299. Search in Google Scholar

[22] C. Qin , Y. Yan , K. Shum , and G. Chen , Mathieu groups and its degree prime-power graphs, Comm. Algebra 47 (2019), no. 10, 4173–4180, https://doi.org/10.1080/00927872.2019.1579342. Search in Google Scholar

[23] C. Qin , Study on the all sporadic simple groups and its irreducible character degree prime-power graphs, Ph.D. Dissertation, Southwest University, Chongqing, 2019. (in Chinese)Search in Google Scholar

Received: 2021-01-04

Revised: 2021-09-19

Accepted: 2021-09-20

Published Online: 2021-12-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2021-0112

Keywords for this article

finite group; character degree; degree prime-power graph; automorphism group

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BY 4.0