Commutators, centralizers, and strong conciseness in profinite groups

Abstract

AbstractA group G is said to have restricted centralizers if for each the centralizer either is finite or has finite index in G. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take interest in profinite groups with restricted centralizers of uniform commutators, that is, elements of the form , where . Here, denotes the set of prime divisors of the order of . It is shown that such a group necessarily has an open nilpotent subgroup. We use this result to deduce that is finite if and only if the cardinality of the set of uniform k‐step commutators in G is less than .