Let G be a finite group. Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_943_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{acd}_{\mathbb {Q}} (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>acd</mtext> <mi mathvariant="double-struck">Q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_943_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{acd}_{\mathbb {Q},2} (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>acd</mtext> <mrow> <mi mathvariant="double-struck">Q</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the average degree of rational irreducible characters of G and, of even degree and linear rational irreducible characters of G, respectively. In this paper, we prove that if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_943_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$ \textrm{acd}_{\mathbb {Q}} (G) < {10}/{3} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>acd</mtext> <mi mathvariant="double-struck">Q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>10</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, then G is solvable. Also, if <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_943_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">$ \textrm{acd}_{\mathbb {Q},2} (G) < {5}/{2} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>acd</mtext> <mrow> <mi mathvariant="double-struck">Q</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mn>5</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, then either G is solvable or some simple group <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_943_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">${\textrm{PSL}}_2(3^{2f+1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>PSL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mrow> <mn>2</mn> <mi>f</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_943_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$f \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) is involved in G.

The average degree of some rational irreducible characters

期刊论文
发表日期：

2025年2月20日

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Esmaeel Eskandari,
Neda Ahanjideh,
Mohsen Ghasemi

摘要

Let G be a finite group. Let $\textrm{acd}_{\mathbb {Q}} (G)$ ${acd}_{Q} (G)$ and $\textrm{acd}_{\mathbb {Q},2} (G)$ ${acd}_{Q, 2} (G)$ be the average degree of rational irreducible characters of G and, of even degree and linear rational irreducible characters of G, respectively. In this paper, we prove that if $ \textrm{acd}_{\mathbb {Q}} (G) < {10}/{3} $ ${acd}_{Q} (G) < 10 / 3$ , then G is solvable. Also, if $ \textrm{acd}_{\mathbb {Q},2} (G) < {5}/{2} $ ${acd}_{Q, 2} (G) < 5 / 2$ , then either G is solvable or some simple group ${\textrm{PSL}}_2(3^{2f+1})$ ${PSL}_{2} (3^{2 f + 1})$ (with $f \ge 1$ $f \geq 1$ ) is involved in G.