z-Classes in groups

Abstract

For a group [Formula: see text], we say that [Formula: see text] are in the same [Formula: see text]-class if their centralizers in [Formula: see text] are conjugate. The notion of [Formula: see text]-class has origin in a connection between geometry and groups. However, as the notion is purely group theoretic, in this paper, we focus our attention on the influence of the [Formula: see text]-classes on the structure of the group. The number of [Formula: see text]-classes is invariant for a family of isoclinic groups. We obtain bounds for the number of [Formula: see text]-classes in certain families of groups. A non-abelian finite [Formula: see text]-group contains at least [Formula: see text] [Formula: see text]-classes. Moreover, we characterize the non-abelian [Formula: see text]-groups with [Formula: see text] [Formula: see text]-classes; these are precisely, up to isoclinism, the [Formula: see text]-groups of maximal class with an abelian subgroup of index [Formula: see text].