Profinite groups with restricted centralizers of powers

A group G is said to have restricted centralizers if for every x∈ G the centralizer CG(x) either is finite or has finite index in G . Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we take interest in profinite groups G for which there is an integer n such that CG(xⁿ) is either finite or open whenever x∈ G . It is shown that such a group G has an open normal subgroup T with the property that G / Z ( T ) has finite exponent.