Bi-skew braces and Hopf Galois structures

摘要

A skew brace G is a set with two group operations, one defining a (not necessarily abelian) "additive group" on G and the other a "circle group" on G, so that G with the two operations satisfies a relation analogous to distributivity. If G is a skew brace, then G yields a Hopf Galois structure of type equal to the additive group of G on any Galois extension of fields with Galois group isomorphic to the circle group of G. A skew brace G is a bi-skew brace if it is also a skew brace with the roles of the circle and additive group reversed. In that event, then G also corresponds to a Hopf Galois structure of type equal to the circle group on a Galois extension of fields with Galois group isomorphic to the additive group. Many non-trivial examples exist. One source is radical rings A with A3 = 0, where one of the groups is abelian and the other need not be. We find that the left braces of degree p3 classified by Bachiller are bi-skew braces if and only they are radical rings. A different source of bi-skew braces is semidirect products of arbitrary finite groups, which yield many examples where both groups are non-abelian, and a skew brace proof of a result of Crespo, Rio and Vela that if G is a semidirect product of two finite groups H and J, then any Galois extension of fields with Galois group G has a Hopf Galois structure of type equal to the direct product of H and J.