Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Abstract

Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 22n+1 and either q − 1, q−2q+1 or q + 2q+1 is a prime number, and M = F4(q), where q = 2n and either q4 + 1 or q4 − q2 + 1 is a prime number. © 2018, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.