Title: Profinite CP-groups

Abstract: Groups all of whose elements have prime power order are called CP-groups. Note that finite groups of this type are precisely the groups in which the centralizer of every nontrivial element has prime power order. Due to classical works of Higman and Suzuki, finite (and locally finite) CP-groups are now fairly well understood.

In this talk we will describe profinite CP-groups. Moreover, we will discuss profinite groups satisfying certain CP-conditions. In particular, we will sketch a proof of the theorem that a profinite group G is virtually pro-p if and only if for every nontrivial element g ∈ G there is a prime p such that the centralizer of g is virtually pro-p.