Conjugacy classes of centralizers in groups

Abstract

Two elements in a group $G$ are said to be in the same $z$-class or $z$-equivalent if their centralizers are conjugate in $G$. This is an equivalence relation on $G$ and provides a partition of $G$ into disjoint equivalence classes. The structure of centralizers and their conjugacy classes provides important insight into the group structure. Although $z$-equivalence is a weaker relation than conjugacy, it is interesting to note that there are in finite groups which have infinitely many conjugacy classes but finitely many $z$-classes. In fact, the finiteness of $z$-classes in algebraic groups and Lie groups is an interesting problem. We have studied the structure of $z$-classes for symmetric groups $S_n$, general linear groups $GL_n(\mathbb{F})$ and general affine groups $GA_n(\mathbb{F})$ and have proven that there are finitely many $z$-classes, for $n \leq$ 5 in $S_n$ and when $\mathbb{F}$ has finitely many extensions, in the latter cases. We also investigate the idea that there is a relation between the finiteness of $z$-classes and and the intuitive understanding of the finiteness of "dynamical types" of transformations in geometry through group actions.