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<p>Let <i>G</i> and <i>A</i> be finite groups of coprime orders, with <i>A</i> acting on <i>G</i> via automorphisms. Within this framework of coprime action, let <i>M</i> be a maximal <i>A</i>-invariant subgroup of <i>G</i>. We define the <i>A</i>-Deskins index of <i>M</i> in <i>G</i> as the order of an <i>A</i>-chief factor <i>H</i>/<i>K</i> of <i>G</i>, where <i>H</i> is an <i>A</i>-invariant minimal normal supplement of <i>M</i> in <i>G</i>. The quotient <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((M \cap H)/K\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>∩</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> is referred to as an <i>AC</i>-section of <i>M</i>. In this paper, we investigate the properties of the <i>A</i>-Deskins index and <i>AC</i>-section of certain maximal <i>A</i>-invariant subgroups in finite groups, and provide several characterizations of the solubility of finite groups, thereby extending previous results in the literature.</p>

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A coprime action version of the Deskins index and c-section of subgroups in finite groups

  • 期刊论文

  • 发表日期：

    2026年3月24日

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  • Yubo Lv,
  • Yangming Li,
  • Xiaoxia Dong

摘要

Let G and A be finite groups of coprime orders, with A acting on G via automorphisms. Within this framework of coprime action, let M be a maximal A-invariant subgroup of G. We define the A-Deskins index of M in G as the order of an A-chief factor H/K of G, where H is an A-invariant minimal normal supplement of M in G. The quotient \((M \cap H)/K\) ( M ∩ H ) / K is referred to as an AC-section of M. In this paper, we investigate the properties of the A-Deskins index and AC-section of certain maximal A-invariant subgroups in finite groups, and provide several characterizations of the solubility of finite groups, thereby extending previous results in the literature.