We determine the structure of the finite non-solvable groups of order divisible by 3 all whose maximal subgroups of order divisible by 3 are supersolvable. Precisely, we demonstrate that if G is a finite non-solvable group satisfying the above condition on maximal subgroups, then either G is a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2443_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$3'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>3</mn> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-group or <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2443_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$G/\textbf{O}_{3'}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">/</mo> <msub> <mi mathvariant="bold">O</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is isomorphic to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2443_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{PSL}_2(2^p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>PSL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>p</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for an odd prime p, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2443_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{O}_{3'}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">O</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denotes the largest normal <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2443_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$3'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>3</mn> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-subgroup of G. Furthermore, in the latter case, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2443_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{O}_{3'}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">O</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is nilpotent and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2443_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{O}_2(G)\le \textbf{Z}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">O</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi mathvariant="bold">Z</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Supersolvable Subgroups of Order Divisible by 3

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2025年6月6日

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Antonio Beltrán,
Changguo Shao

摘要

We determine the structure of the finite non-solvable groups of order divisible by 3 all whose maximal subgroups of order divisible by 3 are supersolvable. Precisely, we demonstrate that if G is a finite non-solvable group satisfying the above condition on maximal subgroups, then either G is a $3'$ $3^{'}$ -group or $G/\textbf{O}_{3'}(G)$ $G / O_{3^{'}} (G)$ is isomorphic to $\textrm{PSL}_2(2^p)$ ${PSL}_{2} (2^{p})$ for an odd prime p, where $\textbf{O}_{3'}(G)$ $O_{3^{'}} (G)$ denotes the largest normal $3'$ $3^{'}$ -subgroup of G. Furthermore, in the latter case, $\textbf{O}_{3'}(G)$ $O_{3^{'}} (G)$ is nilpotent and $\textbf{O}_2(G)\le \textbf{Z}(G)$ $O_{2} (G) \leq Z (G)$ .