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<p>The aim of this paper is to prove the following result: Let <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> be a set of odd primes. If a finite group <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(G=AB\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mi>A</mi> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> is the product of a <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>-subgroup <i>A</i> and a <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>-soluble subgroup <i>B</i>, then the composition factors of <i>G</i> are known. As a consequence, a criterion for such a factorized group to have a normal <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>-subgroup is obtained.</p>

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On the Product of a \(\pi \)-Group and a \(\pi \)-Soluble Group

  • 期刊论文

  • 发表日期：

    2025年11月20日

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  • L. S. Kazarin,
  • A. Martínez-Pastor,
  • M. D. Pérez-Ramos

摘要

The aim of this paper is to prove the following result: Let \(\pi \) π be a set of odd primes. If a finite group \(G=AB\) G = A B is the product of a \(\pi \) π -subgroup A and a \(\pi \) π -soluble subgroup B, then the composition factors of G are known. As a consequence, a criterion for such a factorized group to have a normal \(\pi \) π -subgroup is obtained.