Characterizable and V-characterizable groups

Abstract

Let G be a finite group. The spectrum π_e(G) is the set of all element orders of G. A vanishing element of G is an element g∈G such that χ(g)= 0 for some irreducible complex character χ of G. Denote by Vo(G) the set of the orders of vanishing elements of G. For a set Ω of positive integers, let h(Ω) (v(Ω)) be the number of isomorphism classes of finite group G such that π_e(G)= Ω Vo(G)= Ω, respectively). A group G is called characterizable  (V-characterizable) if h(π_e(G))= 1 (v(Vo(G))= 1, respectively). In this note, we discuss the relation between characterizable and  V-recognizable. Moreover, by an application of the the relation, we prove that the group M₂₂ is V-characterizable.