A new characterization of the automorphism groups of Mathieu groups


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                        <jats:tex-math>{\rm{cd}}\left(G)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> be the set of irreducible complex character degrees of a finite group <jats:inline-formula>
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                        <jats:tex-math>\rho \left(G)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> denotes the set of primes dividing degrees in <jats:inline-formula>
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                        <jats:tex-math>{\rm{cd}}\left(G)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>. For any prime <jats:italic>p</jats:italic>, let <jats:inline-formula>
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                        <jats:tex-math>{p}^{{e}_{p}\left(G)}=\max \left\{\chi {\left(1)}_{p}\hspace{0.08em}| \hspace{0.08em}\chi \in {\rm{Irr}}\left(G)\right\}</jats:tex-math>
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                        <jats:tex-math>V\left(G)=\left\{{p}^{{e}_{p}\left(G)}\hspace{0.08em}| \hspace{0.1em}p\in \rho \left(G)\right\}</jats:tex-math>
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                  </jats:inline-formula>. The degree prime-power graph <jats:inline-formula>
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                  </jats:inline-formula>. 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.</jats:p>

such that xy m | . It is an interesting and difficult problem to determine the structure of a finite group by using its degree primepower 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.

Introduction
The groups involved in this paper are all finite groups, and all characters are complex characters.
We use G Irr( ) to denote the set of all complex irreducible characters of a group G, and G cd( ) = χ χ G 1 Irr to denote the set of all irreducible character degrees of G forgetting multiplicities. In particular, G cd ( ) * denotes a multiple set whose elements can be repeated and . Denote by ρ G ( ) the set of primes dividing degrees in G cd( ). We use G Out( ) 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 n p | , where n is a positive integer and p is a prime. Let n n p π n 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 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 q Sz( ), the alternating groups A n with n 5 1 1 ≤ ≤ , 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 ( ), and q q PSp 7 4 ( ) ( ) > , 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 M cd( ), 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 G Δ( ), is a graph with the vertex set ρ G ( ) and two vertices x and y are adjacent in such that xy f | (see [17]). Many researchers began to investigate the relationship between G Δ( ) and structure of finite group, trying to know about the where q is an odd prime or a square of an odd prime, and q 5 ⩾ ), and L 2 α 2 ( ) (where α is a positive integer such that 2 1 α − or 2 1 α + 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 M 4 1 1 × 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: 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:

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.
If N is an arbitrary non-unit and solvable subnormal subgroup of group G, then N satisfies one of the following conditions: , then one of the following holds: (1) The number of irreducible characters with same degree is less than 11; (2) G has a normal abelian Sylow 2-subgroup.  , and E G 5~11 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 G 1 11 ( ) = . By Lemma 2.3, there is a normal series H K G 1 ⩽ < ⩽ such that K H / is a non-abelian chief factor of G and G K K H Out | | | | ( )| / / . By comparing the order of G and the orders of the simple groups in [1], every nonabelian chief factor of G is isomorphic to one of the following groups: A 5 , A 6 , L 11 2 ( ), M 11 , and M 12 . And the non-abelian chief factors of G are pairwise non-isomorphic. , and so H 2 3 11 α | | = ⋅ ⋅ , where α 2 = , 3, or 4. For the same reasons as above, we also get a contradiction.
Assume that K H M 11 / ≅ , by M 2 3 5 11 11 4 2 | | = ⋅ ⋅ ⋅ , and Out M 1 and Lemma 2.1, there is a character θ H . Therefore, G is a central extension of Z 2 by M 12 and G is isomorphic to one of the following groups:  | | = ⋅ ⋅ ⋅ ⋅ , the non-abelian chief factors of G are pairwise nonisomorphic, and by [1], we see that K H / is isomorphic to one of the following groups: A 5 , L 2 3 ( ), A 6 , L 8 2 ( ), L 11 2 ( ), A 7 , M 11 , A 8 , L 4 3 ( ), 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 N 11|| | by Corollary 2.6. Then N has a normal Sylow 11-subgroup by Lemma 2.2. This contradicts O N 1 11 ( ) = by Lemma 2.1. Hence, G has a chief factor isomorphic to L 11 2 ( ), M 11 , or M 22 .
Assume that G has a chief factor isomorphic to L 11 2 ( ).