A CHARACTERIZATION OF MATHIEU GROUPS BY THEIR ORDERS AND CHARACTER DEGREE GRAPHS

Abstract

Let G be a finite group.  The character degree graph Γ(G) of G is the graph whose vertices are the prime divisors of character degrees of G and two vertices p and q are joined by an edge if pq divides some character degree of G.  Let L_n(q) be the projective special linear group of degree n over finite field of order q.  Xu et al. proved that the Mathieu groups are characterized by the order and one irreducible character degree.  Recently Khosravi et al. have proven that the simple groups L₂(p²), and L₂(p) where p ∈ {7, 8, 11, 13, 17, 19} are characterizable by the degree graphs and their orders.  In this paper, we give a new characterization of Mathieu groups by using the character degree graphs and their orders.