A new characterization of the automorphism groups of Mathieu groups

Let cd(G){\rm{cd}}\left(G) be the set of irreducible complex character degrees of a finite group GG. ρ(G)\rho \left(G) denotes the set of primes dividing degrees in cd(G){\rm{cd}}\left(G). For any prime p, let pep(G)=max{χ(1)p∣χ∈Irr(G)}{p}^{{e}_{p}\left(G)}=\max \left\{\chi {\left(1)}_{p}\hspace{0.08em}| \hspace{0.08em}\chi \in {\rm{Irr}}\left(G)\right\} and V(G)={pep(G)∣p∈ρ(G)}V\left(G)=\left\{{p}^{{e}_{p}\left(G)}\hspace{0.08em}| \hspace{0.1em}p\in \rho \left(G)\right\}. The degree prime-power graph Γ(G)\Gamma \left(G) of GG is a graph whose vertices set is V(G)V\left(G), and two vertices x,y∈V(G)x,y\in V\left(G) are joined by an edge if and only if there exists m∈cd(G)m\in {\rm{cd}}\left(G) such that xy∣mxy| 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.

Abstract licence: CC BY

14 more references, whose wording the paper’s licence keeps private.

A person counts once a day per paper. We keep no cookie and no address: a reader is a hash that changes every day. Any month under five reads shows no number.