Affiliation:
1. Dipartimento di Matematica “Tullio Levi‐Civita” Università di Padova Padova Italy
2. Dipartimento di Matematica Università di Bologna Bologna Italy
3. Department of Mathematics University of Brasilia Brasilia‐DF Brazil
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 .
Cited by
1 articles.
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