论文标题
分裂元环$ p $ -groups的多个群体
The multiple holomorph of split metacyclic $p$-groups
论文作者
论文摘要
给定任何组$ g $，normalizer $ \ mathrm {hol}（g）$的左翻译子组中的所有排列$ g $的左图称为Holomorph，而normanizer $ \ Mathrm {nhol}（nHol}（g）$ of $ \ m m iathrm {h hol}（g）$ n hol holomorph in Chore ny Mulosorph。商$ t（g）= \ mathrm {nhol}（g）/\ mathrm {hol}（g）$已针对文献中各组$ g $的各个家庭计算。在本文中，我们将通过考虑有限的分配Metacyclic $ p $ - groups $ g $，并用$ p $ a奇数来补充现有结果。当$ g $满足某些温和条件时，我们可以为$ t（g）$的订单提供封闭式公式。我们的工作为$ t（g）$不是$ 2 $ group $ g $ $ g $的新家族提供了。
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.
