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<p>Let <i>x</i> be an element of a group <i>G</i>. Denote by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(B_x(G)\)</EquationSource> </InlineEquation> the set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\{a^{-1}x^{-1}a b^{-1}xb \ | \ a,b \in G\}\)</EquationSource> </InlineEquation>. It is well-known that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(B_x(G)\)</EquationSource> </InlineEquation> is a normal subset (of commutators) of <i>G</i>. In this paper we note that some structural results to the group <i>G</i> can be described in terms of restrictions on the subsets <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(B_x(G)\)</EquationSource> </InlineEquation>.</p>

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Finite groups with restrictions on subsets of commutators

  • 期刊论文

  • 发表日期：

    2025年11月10日

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  • T. Araújo de Souza,
  • R. Bastos,
  • E. de Melo

摘要

Let x be an element of a group G. Denote by \(B_x(G)\) the set \(\{a^{-1}x^{-1}a b^{-1}xb \ | \ a,b \in G\}\) . It is well-known that \(B_x(G)\) is a normal subset (of commutators) of G. In this paper we note that some structural results to the group G can be described in terms of restrictions on the subsets \(B_x(G)\) .