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<p>A subgroup <i>L</i> of a finite group <i>G</i> is called a CAP-subgroup of <i>G</i> if for any chief factor <i>H</i>/<i>K</i> of <i>G</i>, we have that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(LH=LK\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mi>H</mi> <mo>=</mo> <mi>L</mi> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L\cap H=L\cap K\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>∩</mo> <mi>H</mi> <mo>=</mo> <mi>L</mi> <mo>∩</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we establish new criteria for the solvability and <i>p</i>-nilpotency of finite groups by analyzing the CAP-subgroups in the context of coprime actions.</p>

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On CAP-Subgroups Under the Coprime Action in Finite Groups

  • 期刊论文

  • 发表日期：

    2025年11月18日

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  • Jianjun Liu,
  • Songtai Li,
  • Xiuyun Guo

摘要

A subgroup L of a finite group G is called a CAP-subgroup of G if for any chief factor H/K of G, we have that \(LH=LK\) L H = L K or \(L\cap H=L\cap K\) L ∩ H = L ∩ K . In this paper, we establish new criteria for the solvability and p-nilpotency of finite groups by analyzing the CAP-subgroups in the context of coprime actions.