收缩映射与有限群中元素阶的倒数和
Deflation map and the sum of inverses of the element orders in finite groups
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中文总结 AI 辅助
本文利用收缩映射在有限群的Burnside环中构造元素，得到元素阶倒数和的表达式，并由此推出交叉Burnside环秩与p-局部Mackey代数Cartan矩阵行列式的另一表达式。
中文摘要 AI 辅助
设G为有限群。本文中，我们利用由G的正规子群N对应的(G/N,G)-双集G/N诱导的收缩映射Def^G_{G/N}，在G在Q上的Burnside环中构造一个元素σ^G，该元素给出有理数m(G)，即G中所有元素阶的倒数之和。作为该定理的推论，我们得到G的交叉Burnside环B^c(G)的秩，以及特征p>0且足够大的域k上G的p-局部Mackey代数的Cartan矩阵行列式的另一表达式。
英文摘要
Let $G$ be a finite group. In this note, we construct an element $σ^G$ in the Burnside ring of $G$ over $\mathbb{Q}$, which gives a rational number $m(G)$ that is the sum of the inverses of the element orders in $G$, by using the deflation map ${\mathrm{Def}}^G_{G/N}$ induced by the $(G/N,G)$-biset $G/N$ for a normal subgroup $N$ of $G$. As a corollary to Theorem, we obtain another expression for the rank of the crossed Burnside ring $B^{\rm c}(G)$ of $G$ and the determinant of the Cartan matrix of $p$-local Mackey algebra of $G$ over a field $k$ of characteristic $p>0$ with big enough.
