Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group

Tsang, Cindy (Sin Yi)

doi:10.1090/bproc/138

Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group
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by Cindy (Sin Yi) Tsang HTML | PDF

Proc. Amer. Math. Soc. Ser. B 9 (2022), 377-392

Abstract:

Let $G$ and $N$ be two finite groups of the same order. It is known that the existences of the following are equivalent.

a Hopf-Galois structure of type $N$ on any Galois $G$-extension
a skew brace with additive group $N$ and multiplicative group $G$
a regular subgroup isomorphic to $G$ in the holomorph of $N$

We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete characterization of the groups $N$ for which $(G,N)$ is realizable.

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