论文标题
在有限可溶组的群体中的不溶性及物子群上
On Insoluble Transitive Subgroups in the Holomorph of a Finite Soluble Group
论文作者
论文摘要
Hopf-Galois理论和偏斜理论中的一个感兴趣问题是，有限可溶组$ n $的Holomorph $ \ Mathrm {hol（n）} $是否可以包含一个不溶性的常规子组。我们研究了一个更普遍的问题，即在$ \ mathrm {hol}（n）$中找到一个不溶性的及物子组$ g $，并使用可溶的点稳定器。如果我们不能通过适当的非无聊的商$ \ Overline {g} $，$ \ overline {n} $ of $ g $，$ n $，以便$ \ overline {g <g} $，我们称之为一对$（g，n）$不可记述。我们对此问题的所有不可约（G，n）$进行了分类，特别表明，每个非亚伯式的成分因子$ g $均为$ g $的$ 168 $ 168 $。此外，$ n $的每个最大正常子组的指数$ 2 $。
A question of interest both in Hopf-Galois theory and in the theory of skew braces is whether the holomorph $\mathrm{Hol(N)}$ of a finite soluble group $N$ can contain an insoluble regular subgroup. We investigate the more general problem of finding an insoluble transitive subgroup $G$ in $\mathrm{Hol}(N)$ with soluble point stabilisers. We call such a pair $(G,N)$ irreducible if we cannot pass to proper non-trivial quotients $\overline{G}$, $\overline{N}$ of $G$, $N$ so that $\overline{G}$ becomes a subgroup of $\mathrm{Hol}(\overline{N})$. We classify all irreducible solutions $(G,N)$ of this problem, showing in particular that every non-abelian composition factor of $G$ is isomorphic to the simple group of order $168$. Moreover, every maximal normal subgroup of $N$ has index $2$.
