Profinite groups with restricted centralizers of powers

Abstract A group G is said to have restricted centralizers if for every $$x\in G$$ the centralizer $$C_G(x)$$ 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 $$C_G(x^n)$$ is either finite or open whenever $$x\in G$$ . It is shown that such a group G has an open normal subgroup T with the property that G / Z ( T ) has finite exponent.