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arXiv 2004.070842020-04math.GR

The multiple holomorph of split metacyclic $p$-groups

Cindy Tsang

备注：: very minor changes
发表信息：: Comm. Algebra 50 (2022), no. 10, 4269-4287
DOI：: 10.1080/00927872.2022.2059079

英文摘要：

Given any group $G$, the normalizer $\mathrm{Hol}(G)$ of the subgroup of left translations in the group of all permutations on $G$ is called the holomorph, and the normalizer $\mathrm{NHol}(G)$ of $\mathrm{Hol}(G)$ in turn is called the multiple holomorph. The quotient $T(G) = \mathrm{NHol}(G)/\mathrm{Hol}(G)$ has been computed for various families of groups $G$ in the literature. In this paper, we shall supplement the existing results by considering finite split metacyclic $p$-groups $G $ with $p$ an odd prime. We are able to give a closed formula for the order of $T(G)$ when $G$ satisfies some mild conditions. Our work gives a new family of groups $G$ for which $T(G)$ is not a $2$-group.

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