Profinite groups in which centralizers are virtually procyclic

Abstract The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic ( Theorem 1.1 ). The same conclusion holds for profinite groups in which the centralizer of every nontrivial element is virtually procyclic ( Theorem 1.2 ); moreover, if G is not pro-p, then G has finite rank.