Central intersections of element centralizers

Abstract

In 1970, R. Schmidt gave a structural classification for CA-groups. In this paper, we consider a condition upon the intersection of element centralizers which turns out to be equivalent to the definition of a CA-group. We then weaken which centralizers we chose to intersect and structurally classify this new family of groups. Furthermore, we apply a similar weakening to the class of F-groups introduced by Itô in 1953 and classified by Rebmann in 1971.