Let G be a finite group. A subgroup H of G is said to be an <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_933_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$IC\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mi>C</mi> <mi mathvariant="normal">Φ</mi> </mrow> </math></EquationSource> </InlineEquation>-subgroup of G if <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_933_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="142" /> </InlineMediaObject> <EquationSource Format="TEX">$H\cap [H,G]\le \Phi (H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>∩</mo> <mo stretchy="false">[</mo> <mi>H</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">]</mo> <mo>≤</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we fix a subgroup D of Sylow p-subgroup P of G with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_933_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$1<|D|<|O_p(G)|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mn>1</mn> <mo><</mo> <mo stretchy="false">|</mo> <mi>D</mi> <mo stretchy="false">|</mo> <mo><</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>O</mi> <mi>p</mi> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and study the structure of G under the assumption that all subgroups H of P with order <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_933_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$|H|=|D|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>H</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mi>D</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> are <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_933_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$IC\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mi>C</mi> <mi mathvariant="normal">Φ</mi> </mrow> </math></EquationSource> </InlineEquation>-subgroups of G.

Finite groups with given $IC\Phi $-subgroups

期刊论文
发表日期：

2025年2月9日

查看全文

Xuanli He,
Xinfang Zhang,
Wenyu Zhu,
Ju Wang

摘要

Let G be a finite group. A subgroup H of G is said to be an $IC\Phi $ $I C Φ$ -subgroup of G if $H\cap [H,G]\le \Phi (H)$ $H \cap [H, G] \leq Φ (H)$ . In this paper, we fix a subgroup D of Sylow p-subgroup P of G with $1<|D|<|O_p(G)|$ $1 < | D | < | O_{p} (G) |$ and study the structure of G under the assumption that all subgroups H of P with order $|H|=|D|$ $| H | = | D |$ are $IC\Phi $ $I C Φ$ -subgroups of G.

Finite groups with given \(IC\Phi \)-subgroups

摘要