Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma = \{ {\sigma }_{i} \mid i \in I \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>=</mo> <mo stretchy="false">{</mo> <msub> <mi>σ</mi> <mi>i</mi> </msub> <mo>∣</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> be a partition of the set of all primes. A set <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="135" /> </InlineMediaObject> <EquationSource Format="TEX">$\Sigma = \{H_1, \ldots , H_k\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <mo stretchy="false">{</mo> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>-Hall subgroups of G is said to be a complete Hall set of type <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> of G if <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">$(|H_i|, |H_j|) = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <msub> <mi>H</mi> <mi>i</mi> </msub> <mo stretchy="false">|</mo> <mo>,</mo> <mo stretchy="false">|</mo> <msub> <mi>H</mi> <mi>j</mi> </msub> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq9.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$i \ne j$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="204" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi (G) = \pi (H_1) \cup \cdots \cup \pi (H_k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>π</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∪</mo> <mo>⋯</mo> <mo>∪</mo> <mi>π</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. A <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-subnormality criterion is proved in terms of products of subgroups of a complete Hall set of type <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2072_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>.

An arithmetic criterion for $\sigma $-subnormality in finite groups

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2025年3月25日

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A. Ballester-Bolinches,
S. F. Kamornikov,
V. Pérez-Calabuig,
O. L. Shemetkova

摘要

Let $\sigma = \{ {\sigma }_{i} \mid i \in I \}$ $σ = {σ_{i} ∣ i \in I}$ be a partition of the set of all primes. A set $\Sigma = \{H_1, \ldots , H_k\}$ $Σ = {H_{1}, \dots, H_{k}}$ of $\sigma _i$ $σ_{i}$ -Hall subgroups of G is said to be a complete Hall set of type $\sigma $ $σ$ of G if $(|H_i|, |H_j|) = 1$ $(| H_{i} |, | H_{j} |) = 1$ for all $i \ne j$ $i \neq j$ and $\pi (G) = \pi (H_1) \cup \cdots \cup \pi (H_k)$ $π (G) = π (H_{1}) \cup \dots \cup π (H_{k})$ . A $\sigma $ $σ$ -subnormality criterion is proved in terms of products of subgroups of a complete Hall set of type $\sigma $ $σ$ .

An arithmetic criterion for \(\sigma \)-subnormality in finite groups

摘要