Let G be a finite group, and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation> be an irreducible complex character of G. In this note, we define <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{zk}(G)=|\{ \chi \in \textrm{Irr}(G) \bigm |\textrm{Z}(\chi )=\textrm{Ker}(\chi ) \}|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>zk</mtext> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mo stretchy="false">{</mo> <mi>χ</mi> <mo>∈</mo> <mtext>Irr</mtext> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">|</mo> </mrow> <mtext>Z</mtext> <mo stretchy="false">(</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext>Ker</mtext> <mo stretchy="false">(</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>. We prove that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{zk}(G)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>zk</mtext> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> if and only if G is nilpotent, which provides a new criterion for nilpotency. We also show that G is solvable when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{zk}(G)\leqslant 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>zk</mtext> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>⩽</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and characterize non-solvable groups with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{zk}(G)=5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>zk</mtext> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>.

Quasikernels and kernels of irreducible characters

期刊论文
发表日期：

2026年4月16日

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Pujin Li,
Haipeng Qu

摘要

Let G be a finite group, and $\chi $ $χ$ be an irreducible complex character of G. In this note, we define $\textrm{zk}(G)=|\{ \chi \in \textrm{Irr}(G) \bigm |\textrm{Z}(\chi )=\textrm{Ker}(\chi ) \}|$ $zk (G) = | {χ \in Irr (G) | Z (χ) = Ker (χ)} |$ . We prove that $\textrm{zk}(G)=1$ $zk (G) = 1$ if and only if G is nilpotent, which provides a new criterion for nilpotency. We also show that G is solvable when $\textrm{zk}(G)\leqslant 4$ $zk (G) ⩽ 4$ and characterize non-solvable groups with $\textrm{zk}(G)=5$ $zk (G) = 5$ .