A new characterization of the automorphism groups of Mathieu groups

Let be the set of irreducible complex character degrees of a finite group . denotes the set of primes dividing degrees in . For any prime , let and . The degree prime-power graph of is a graph whose vertices set is , and two vertices are joined by an edge if and only if there exists such that . 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.