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<p>Suppose that <i>G</i> is a finite solvable group and let <i>H</i> be a Hall <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>-subgroup, let <i>b</i>(<i>H</i>) be the largest character degree of <i>H</i>, we show that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(|G:O_{\pi ' \pi }(G)|_{\pi } \le b(H)^2.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>G</mi> <mo>:</mo> </mrow> <msub> <mi>O</mi> <mrow> <msup> <mi>π</mi> <mo>′</mo> </msup> <mi>π</mi> </mrow> </msub> <msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mi>π</mi> </msub> <mo>≤</mo> <mi>b</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Large orbits of Hall subgroups of solvable linear groups

  • 期刊论文

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  • 发表日期：

    2026年6月29日

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  • Samarth Das,
  • Yong Yang

摘要

Suppose that G is a finite solvable group and let H be a Hall \(\pi \) π -subgroup, let b(H) be the largest character degree of H, we show that \(|G:O_{\pi ' \pi }(G)|_{\pi } \le b(H)^2.\) | G : O π ′ π ( G ) | π ≤ b ( H ) 2 .