Profinite groups with restricted centralizers of $$\pi $$-elements

Abstract

A group G is said to have restricted centralizers if for each g in G the centralizer \(C_G(g)\) either is finite or has finite index in G. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes \(\pi \), we take interest in profinite groups with restricted centralizers of \(\pi \)-elements. It is shown that such a profinite group has an open subgroup of the form \(P\times Q\), where P is an abelian pro-\(\pi \) subgroup and Q is a pro-\(\pi '\) subgroup. This significantly strengthens a result from our earlier paper.