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 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. [ABSTRACT FROM AUTHOR]