Let G be a finite group, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> be a set of prime numbers. In this paper, we introduce an invertible integer matrix <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_{\pi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>π</mi> </msub> </math></EquationSource> </InlineEquation> which depends on the pair <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(G,\pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This matrix generalizes the Cartan matrix of G for a prime p. We prove that for a nilpotent group G, the largest elementary divisor of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_{\pi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>π</mi> </msub> </math></EquationSource> </InlineEquation> is equal to the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>-part <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$|G|_{\pi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">|</mo> </mrow> <mi>π</mi> </msub> </math></EquationSource> </InlineEquation> of |G|. Furthermore, we consider a matrix <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$W_{\pi }(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> obtained by restricting the character table of G to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$G_{\pi '}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <msup> <mi>π</mi> <mo>′</mo> </msup> </msub> </math></EquationSource> </InlineEquation> and then rationalizing it. Then we prove that for any finite group G, the largest elementary divisor of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$W_{\pi }(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>π</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is equal to the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\pi '$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>π</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-part <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$|G|_{\pi '}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">|</mo> </mrow> <msup> <mi>π</mi> <mo>′</mo> </msup> </msub> </math></EquationSource> </InlineEquation> of |G|.

Elementary divisors of matrices obtained from the character table of finite groups

期刊论文
发表日期：

2026年8月1日

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Nobuo Iiyori,
Masao Kiyota,
Masato Sawabe

摘要

Let G be a finite group, and let $\pi $ $π$ be a set of prime numbers. In this paper, we introduce an invertible integer matrix $C_{\pi }$ $C_{π}$ which depends on the pair $(G,\pi )$ $(G, π)$ . This matrix generalizes the Cartan matrix of G for a prime p. We prove that for a nilpotent group G, the largest elementary divisor of $C_{\pi }$ $C_{π}$ is equal to the $\pi $ $π$ -part $|G|_{\pi }$ ${| G |}_{π}$ of |G|. Furthermore, we consider a matrix $W_{\pi }(G)$ $W_{π} (G)$ obtained by restricting the character table of G to $G_{\pi '}$ $G_{π^{'}}$ and then rationalizing it. Then we prove that for any finite group G, the largest elementary divisor of $W_{\pi }(G)$ $W_{π} (G)$ is equal to the $\pi '$ $π^{'}$ -part $|G|_{\pi '}$ ${| G |}_{π^{'}}$ of |G|.