关于有限几乎单群的重全形
On the multiple holomorph of a finite almost simple group
摘要：
设$G$是一个群。令$\mathrm{Perm}(G)$表示其对称群，并将$\mathrm{Perm}(G)$中左平移子群的正规化子写作$\mathrm{Hol}(G)$。$G$的多重全像$\mathrm{NHol}(G)$又被定义为$\mathrm{Perm}(G)$中$\mathrm{Hol}(G)$的正规化子。本文证明了当$G$有限且几乎单时，商群$\mathrm{NHol}(G)/\mathrm{Hol}(G)$的阶为2.作为我们的技巧的应用，我们还将发展出一种方法来计算有限几乎单扩张上同构型的Hopf-Galois结构的个数.
英文摘要：
Let $G$ be a group. Let $\mathrm{Perm}(G)$ denote its symmetric group and write $\mathrm{Hol}(G)$ for the normalizer of the subgroup of left translations in $\mathrm{Perm}(G)$. The multiple holomorph $\mathrm{NHol}(G)$ of $G$ is in turn defined to be the normalizer of $\mathrm{Hol}(G)$ in $\mathrm{Perm}(G)$. In this paper, we shall show that the quotient group $\mathrm{NHol}(G)/\mathrm{Hol}(G)$ has order two when $G$ is finite and almost simple. As an application of our techniques, we shall also develop a method to count the number of Hopf-Galois structures of isomorphic type on a finite almost simple extension in terms of fixed point free endomorphisms.
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