Abstract 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.
Acciarri et al. (Sat,) studied this question.