Profinite groups with restricted centralizers of π -elements

A group G is said to have restricted centralizers if for each g in G the centralizer CG(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.