Partial GVZ-groups

Abstract

Following the literature, a group G is called a group of central type if G has an irreducible character that vanishes on G \ Z(G). Motivated by this definition, we say that a character χ ∈ Irr(G) has central type if χ vanishes on G \ Z(χ), where Z(χ) is the center of χ. Groups where every irreducible character has central type have been studied previously under the name GVZ-groups (and several other names) in the literature. In this paper, we study the groups G that possess a nontrivial, normal subgroup N such that every character of G either contains N in its kernel or has central type. The structure of these groups is surprisingly limited and has many aspects in common with both central type groups and GVZ-groups.