Flatness Conditions on Finite p -Groups#

Abstract

A group is flat if every finite conjugacy class is a coset of a (necessarily normal) subgroup. This paper is the first to consider the consequences of flatness condition for finite p-groups. The flatness condition is weaker than that of Camina pairs (as we shall prove in this paper) and a key aim of this paper is to prove a similar main result to that in the theory of such pairs. Consequences for orders, nilpotence classes and orders of centre of the groups are deduced.