Let G be a finite group, and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2401_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> be a set of primes. The <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2401_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>-core <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2401_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{O}_\pi (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">O</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the unique maximal normal <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2401_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>-subgroup of G, and b(G) is the largest irreducible character degree of G. In 2017, Qian and Yang proved that if H is a solvable <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2401_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>-subgroup of G, then <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2401_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="191" /> </InlineMediaObject> <EquationSource Format="TEX">$|H\textbf{O}_\pi (G)/\textbf{O}_\pi (G)|\le b(G)^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>H</mi> </mrow> <msub> <mi mathvariant="bold">O</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mi mathvariant="bold">O</mi> <mi>π</mi> </msub> <msup> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we improve the exponent of 3 to <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2401_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="146" /> </InlineMediaObject> <EquationSource Format="TEX">$3\log _{504}(168)<2.471$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <msub> <mo>log</mo> <mn>504</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>168</mn> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>2.471</mn> </mrow> </math></EquationSource> </InlineEquation>. Along the way, we also prove that if a solvable group P acts faithfully on X, then there is some 3-coloring of X such that there are no more than <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2401_Article_IEq8.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\sqrt{|P|}$</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mrow> <mo stretchy="false">|</mo> <mi>P</mi> <mo stretchy="false">|</mo> </mrow> </msqrt> </math></EquationSource> </InlineEquation> elements of P that preserve the coloring.

On the Largest Character Degree and Solvable Subgroups of Finite Groups

期刊论文
发表日期：

2025年4月10日

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Zongshu Wu,
Yong Yang

摘要

Let G be a finite group, and $\pi $ $π$ be a set of primes. The $\pi $ $π$ -core $\textbf{O}_\pi (G)$ $O_{π} (G)$ is the unique maximal normal $\pi $ $π$ -subgroup of G, and b(G) is the largest irreducible character degree of G. In 2017, Qian and Yang proved that if H is a solvable $\pi $ $π$ -subgroup of G, then $|H\textbf{O}_\pi (G)/\textbf{O}_\pi (G)|\le b(G)^3$ $| H O_{π} (G) / O_{π} {(G) | \leq b (G)}^{3}$ . In this paper, we improve the exponent of 3 to $3\log _{504}(168)<2.471$ $3 {log}_{504} (168) < 2.471$ . Along the way, we also prove that if a solvable group P acts faithfully on X, then there is some 3-coloring of X such that there are no more than $\sqrt{|P|}$ $\sqrt{| P |}$ elements of P that preserve the coloring.