Let H be a subgroup of a finite group G. We say that H satisfies the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \Pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Π</mi> </math></EquationSource> </InlineEquation>-property in G if for any chief factor L/K of G, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ |G/K: N_{G/K}(HK/K\cap L/K )| $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo>:</mo> </mrow> <msub> <mi>N</mi> <mrow> <mi>G</mi> <mo stretchy="false">/</mo> <mi>K</mi> </mrow> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mi>K</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo>∩</mo> <mi>L</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ \pi (HK/K\cap L/K) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mi>K</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo>∩</mo> <mi>L</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-number. In this paper, we obtain a criterion of a normal subgroup being contained in the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ p\mathfrak {F} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mi mathvariant="fraktur">F</mi> </mrow> </math></EquationSource> </InlineEquation>-hypercenter of a finite group by the <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ \Pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Π</mi> </math></EquationSource> </InlineEquation>-property of some p-subgroups, which improves some known results.

On the $ \Pi $-property of subgroups and the $ p\mathfrak {F} $-hypercenter of a finite group

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2026年3月13日

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Zhengtian Qiu,
ShouHong Qiao

摘要

Let H be a subgroup of a finite group G. We say that H satisfies the $ \Pi $ $Π$ -property in G if for any chief factor L/K of G, $ |G/K: N_{G/K}(HK/K\cap L/K )| $ $| G / K : N_{G / K} (H K / K \cap L / K) |$ is a $ \pi (HK/K\cap L/K) $ $π (H K / K \cap L / K)$ -number. In this paper, we obtain a criterion of a normal subgroup being contained in the $ p\mathfrak {F} $ $p F$ -hypercenter of a finite group by the $ \Pi $ $Π$ -property of some p-subgroups, which improves some known results.

On the \( \Pi \)-property of subgroups and the \( p\mathfrak {F} \)-hypercenter of a finite group

摘要