Profinite groups in which many elements have prime power order

We show that a profinite group G is virtually pro-p for some prime p if and only if for each nontrivial x is an element of G there is a prime p (depending on x) such that C-G (x) is virtually pro-p. Further, if G is a profinite group in which each element has either finite or prime power (possibly infinite) order, then G is either torsion or virtually pro-p for some prime p. A detailed description of profinite groups in which every element has prime power order is also given. (C) 2020 Elsevier Inc. All rights reserved.