Profinite groups with restricted centralizers of $π$-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 π, we take interest in profinite groups with restricted centralizers of π-elements. It is shown that such a profinite group has an open subgroup of the form P × Q, where P is an abelian pro-π subgroup and Q is a pro-π subgroup. This significantly strengthens a result from our earlier paper.2010 Mathematics Subject Classification. 20E18, 20F24.