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Abstract <p> We study <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(D_\pi\)</EquationSource> </InlineEquation>-groups with a trivial solvable radical that do not have nontrivial normal <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\pi\)</EquationSource> </InlineEquation>-subgroups in which all simple nonabelian factors of their subnormal series are simple sporadic groups. It is proved that in such groups, for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\pi\)</EquationSource> </InlineEquation>-Hall subgroup <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(H\)</EquationSource> </InlineEquation>, there exists an element <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(g\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(H\cap H^g=1\)</EquationSource> </InlineEquation>. Thus, Problem 20.123(c) of the Kourovka Notebook is solved and, under the above conditions, a positive answer is given to Problem 18.31. </p>

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On Intersections of \(\pi\)-Hall Subgroups of Some \(D_\pi\)-Groups

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • I. N. Belousov,
  • V. I. Zenkov

摘要

Abstract

We study \(D_\pi\) -groups with a trivial solvable radical that do not have nontrivial normal \(\pi\) -subgroups in which all simple nonabelian factors of their subnormal series are simple sporadic groups. It is proved that in such groups, for any \(\pi\) -Hall subgroup \(H\) , there exists an element \(g\) such that \(H\cap H^g=1\) . Thus, Problem 20.123(c) of the Kourovka Notebook is solved and, under the above conditions, a positive answer is given to Problem 18.31.