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2019•Proceedings of the Royal Society of Edinburgh Section A MathematicsOpen access

Profinite groups with restricted centralizers of commutators

Eloisa Detomi, Marta Morigi, Pavel Shumyatsky

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Abstract

Abstract A groupGhas restricted centralizers if for eachginGthe centralizer $C_G(g)$ either is finite or has finite index inG. 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 ifwis a multilinear commutator word andGa profinite group with restricted centralizers ofw-values, then the verbal subgroupw(G) is abelian-by-finite.

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Abstract A groupGhas restricted centralizers if for eachginGthe centralizer $C_G(g)$ either is finite or has finite index inG. 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 ifwis a multilinear commutator word andGa profinite group with restricted centralizers ofw-values, then the verbal subgroupw(G) is abelian-by-finite.

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Available abstract

Abstract A groupGhas restricted centralizers if for eachginGthe centralizer $C_G(g)$ either is finite or has finite index inG. 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 ifwis a multilinear commutator word andGa profinite group with restricted centralizers ofw-values, then the verbal subgroupw(G) is abelian-by-finite.

Key concepts: Profinite group, Centralizer and normalizer, Commutator, Multilinear map, Mathematics, Abelian group, Pure mathematics, Word (group theory)

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