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<p>Let <i>A</i> be a finite non-metacyclic group, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(A^\#\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>A</mi> <mo>#</mo> </msup> </math></EquationSource> </InlineEquation> denote the set of non-trivial elements of <i>A</i>. Suppose that <i>A</i> acts coprimely on a finite group <i>G</i> in such a manner that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C_G(a)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is nilpotent for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(a\in A^{\#}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <msup> <mi>A</mi> <mo>#</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. In the present paper we investigate some conditions on <i>A</i> which imply that <i>G</i> is nilpotent with “bounded” nilpotency class. More precisely, we generalize known results on action of <i>q</i>-groups and Frobenius groups, where <i>q</i> is a prime.</p>

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Non-metacyclic Groups Acting with Nilpotent Centralizers

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Emerson de Melo,
  • Jhone Caldeira

摘要

Let A be a finite non-metacyclic group, and let \(A^\#\) A # denote the set of non-trivial elements of A. Suppose that A acts coprimely on a finite group G in such a manner that \(C_G(a)\) C G ( a ) is nilpotent for any \(a\in A^{\#}\) a ∈ A # . In the present paper we investigate some conditions on A which imply that G is nilpotent with “bounded” nilpotency class. More precisely, we generalize known results on action of q-groups and Frobenius groups, where q is a prime.