Hopf-Galois realizability of Z(n) (sic) Z(2)

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 Hol(N) = N (sic) Aut(N). In this article we give necessary conditions on G (similarly N) when N (similarly G) is a group of the form Z(n) (sic) Z(2), for (G, N) to be realizable. Further we show that this condition is also sufficient if radical of n is a Burnside number. This classifies all skew braces which have the additive group (or the multiplicative group) isomorphic to Zn A Z2, in this case