满足不等式条件的直积的最小忠实置换度
The Smallest Faithful Permutation Degree for a Direct Product Obeying an Inequality Condition
摘要：
有限群$G$的最小忠实置换度$μ(G)$是使$G$嵌入对称群$\Sym(n)$的最小非负整数$n$。显然$μ(G \times H) \le μ(G) + μ(H)$对所有有限群$G$和$H$。Wright（1975）证明了当$G$和$H$是幂零的时等式成立，并给出了一个严格不等式的例子，其中$G\times H$嵌入$\Sym(15)$。Saunders（2010）给出了置换群$G$和$H$的无限族例子，其中$μ(G \times H) < μ(G) + μ(H)$，包括Wright的例子作为特例。Saunders类中的最小群嵌入在$\Sym(10)$中。在本文中，我们证明了10是最小的意义下，$μ(G \times H) = μ(G) + μ(H)$对所有的群$G$和$H$，使得$μ(G\times H)\le 9$。
英文摘要：
The minimal faithful permutation degree $μ(G)$ of a finite group $G$ is the least nonnegative integer $n$ such that $G$ embeds in the symmetric group $\Sym(n)$. Clearly $μ(G \times H) \le μ(G) + μ(H)$ for all finite groups $G$ and $H$. Wright (1975) proves that equality occurs when $G$ and $H$ are nilpotent and exhibits an example of strict inequality where $G\times H$ embeds in $\Sym(15)$. Saunders (2010) produces an infinite family of examples of permutation groups $G$ and $H$ where $μ(G \times H) < μ(G) + μ(H)$, including the example of Wright's as a special case. The smallest groups in Saunders' class embed in $\Sym(10)$. In this paper we prove that 10 is minimal in the sense that $μ(G \times H) = μ(G) + μ(H)$ for all groups $G$ and $H$ such that $μ(G\times H)\le 9$.
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