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.