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Profinite groups with restricted centralizers of commutators

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  01 April 2019

Eloisa Detomi ,
Marta Morigi  and
Pavel Shumyatsky
Eloisa Detomi
Affiliation:
Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121Padova, Italy (detomi@math.unipd.it)
Marta Morigi
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126Bologna, Italy (marta.morigi@unibo.it)
Pavel Shumyatsky
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900Brazil (pavel@unb.br)
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Abstract

A group G has restricted centralizers if for each g in G the centralizer $C_G(g)$ either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present paper we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.

Keywords

group wordsprofinite groupscentralizersFC-groups

MSC classification

Primary: 20F24: FC-groups and their generalizations
Secondary: 20E18: Limits, profinite groups 20F12: Commutator calculus
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2301 - 2321
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Copyright
Copyright © Royal Society of Edinburgh 2019

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Footnotes

Dedicated to Aner Shalev on the occasion of his 60th birthday.

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