Hopf-Galois Realizability of ℤ_n\rtimesℤ₂

2022/01/26 by Namrata Arvind, Arvind, Namrata, Saikat Panja +1
Mathematics · #12F10 #16T05 #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2201.10862

openalex publication_date 2022/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ℤ_n\rtimesℤ₂. 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 ℤ_n\rtimesℤ₂, in this case.

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