z-classes in groups: a survey

Abstract

This survey article explores the notion of z-classes in groups. The concept introduced here is related to the notion of orbit types in transformation groups, and types or genus in the representation theory of finite groups of Lie type. Two elements in a group are said to be z-equivalent (or z-conjugate) if their centralizers are conjugate. This is a weaker notion than the conjugacy of elements. In this survey article, we present several known results on this topic and suggest some further questions.