Abstract <p> Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sigma = \{\sigma_i \mid i \in I \}\)</EquationSource> </InlineEquation> be a partition of the set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb{P}\)</EquationSource> </InlineEquation> of all primes, and let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> be a finite group. A set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> </InlineEquation> of subgroups of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> is called a complete Hall <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\sigma\)</EquationSource> </InlineEquation>-set of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> if every subgroup in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> </InlineEquation> is a <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\sigma_i\)</EquationSource> </InlineEquation>-Hall subgroup of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> for every <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(i \in I\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> </InlineEquation> contains exactly one <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(\sigma_i\)</EquationSource> </InlineEquation>-Hall subgroup for every <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(i\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(\sigma_i \cap \pi (G) \neq \varnothing\)</EquationSource> </InlineEquation>. In this paper, we study the structure of the group <InlineEquation ID="IEq17"> <EquationSource Format="TEX">\(G \in \bigcap_{i \in I}D_{\sigma_i}(\mathfrak {S})\)</EquationSource> </InlineEquation> under the condition that all subgroups in every complete Hall <InlineEquation ID="IEq18"> <EquationSource Format="TEX">\(\sigma\)</EquationSource> </InlineEquation>-set of the group <InlineEquation ID="IEq19"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> are permutable. </p>

Finite Groups with a Solvable Hall \(\sigma\)-Basis

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发表日期：

2025年12月1日

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S. F. Kamornikov,
V. N. Tyutyanov,
O. L. Shemetkova

摘要

Abstract

Let \(\sigma = \{\sigma_i \mid i \in I \}\) be a partition of the set \(\mathbb{P}\) of all primes, and let \(G\) be a finite group. A set \(\mathcal {H}\) of subgroups of \(G\) is called a complete Hall \(\sigma\) -set of \(G\) if every subgroup in \(\mathcal {H}\) is a \(\sigma_i\) -Hall subgroup of \(G\) for every \(i \in I\) and \(\mathcal {H}\) contains exactly one \(\sigma_i\) -Hall subgroup for every \(i\) such that \(\sigma_i \cap \pi (G) \neq \varnothing\) . In this paper, we study the structure of the group \(G \in \bigcap_{i \in I}D_{\sigma_i}(\mathfrak {S})\) under the condition that all subgroups in every complete Hall \(\sigma\) -set of the group \(G\) are permutable.