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<p>Given a prime <i>p</i> and a subgroup <i>A</i> of a finite group <i>G</i>, we say that <i>A</i> is a <i>p</i>-CAP-subgroup of <i>G</i> if <i>A</i> covers or avoids every <i>p-G</i>-chief factor, where a <i>p-G</i>-chief factor is a <i>G</i>-chief factor of order divisible by <i>p</i>. We say that <i>A</i> is a strong <i>p</i>-CAP-subgroup of <i>G</i> if <i>A</i> is a <i>p</i>-CAP-subgroup of any subgroup of <i>G</i> containing <i>A</i>. We use the concept of strong <i>p</i>-CAP-subgroups to investigate the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p\mathfrak{F}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>p</mi> <mrow> <mi mathvariant="fraktur">F</mi> </mrow> </math></EquationSource> </InlineEquation>-hypercentrally embedded property of normal subgroups of a finite group and obtain some new results. Moreover, we extend the concept of (strong) <i>p</i>-CAP-subgroups to fusion systems and use this to characterize supersolvable and nilpotent fusion systems.</p>

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Hypercyclically embedded property and supersolvable fusion systems

  • 期刊论文

  • 发表日期：

    2026年5月11日

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  • Yaxin Gao,
  • Julian Kaspczyk

摘要

Given a prime p and a subgroup A of a finite group G, we say that A is a p-CAP-subgroup of G if A covers or avoids every p-G-chief factor, where a p-G-chief factor is a G-chief factor of order divisible by p. We say that A is a strong p-CAP-subgroup of G if A is a p-CAP-subgroup of any subgroup of G containing A. We use the concept of strong p-CAP-subgroups to investigate the \(p\mathfrak{F}\) p F -hypercentrally embedded property of normal subgroups of a finite group and obtain some new results. Moreover, we extend the concept of (strong) p-CAP-subgroups to fusion systems and use this to characterize supersolvable and nilpotent fusion systems.