论文标题
$ \ mathbb {z} _n \ rtimes \ mathbb {z} _2 $
Hopf-Galois Realizability of $\mathbb{Z}_n\rtimes\mathbb{Z}_2$
论文作者
论文摘要
令$ g $和$ n $是有限群，订单$ 2N $，其中$ n $很奇怪。我们说，如果$ g $是$ \ h（n）= n \ rtimes \ au（n）$的常规子组，那么这对$（g，n）$是可以实现的hopf-galois。在本文中，当$ n $（类似的$ g $）是一组$ \ mathbb {z} _n \ rtimes \ mathbb {z} _2 $时，我们给出了$ g $（类似$ n $）的必要条件。此外，我们表明，如果$ n $的根本是一个伯恩赛德号，则这种情况也足够了。这将分类所有具有同型组（或乘法组）为$ \ Mathbb {z} _n \ rtimes \ mathbb {z} _2 $的偏斜括号。
Let $G$ and $N$ be finite groups of order $2n$ where $n$ is odd. We say the pair $(G,N)$ is Hopf-Galois realizable if $G$ is a regular subgroup of $\h(N)=N\rtimes\au(N)$. In this article we give necessary conditions on $G$ (similarly $N$) when $N$ (similarly $G$) is a group of the form $\mathbb{Z}_n\rtimes\mathbb{Z}_2$. Further we show that this condition is also sufficient if radical of $n$ is a Burnside number. This classifies all the skew braces which has the additive group (or the multiplicative group) to be isomorphic to $\mathbb{Z}_n\rtimes\mathbb{Z}_2$, in this case.
