On profinite groups in which centralizers have bounded rank

Abstract

The paper deals with profinite groups in which centralizers are of finite rank. For a positive integer [Formula: see text] we prove that if [Formula: see text] is a profinite group in which the centralizer of every nontrivial element has rank at most [Formula: see text], then [Formula: see text] is either a pro-[Formula: see text] group or a group of finite rank. Further, if [Formula: see text] is not virtually a pro-[Formula: see text] group, then [Formula: see text] is virtually of rank at most [Formula: see text].