Non-existence of Hopf–Galois structures and bijective crossed homomorphisms

Abstract

By work of C. Greither and B. Pareigis as well as N. P. Byott, the enumeration of Hopf–Galois structures on a Galois extension of fields with Galois group G may be reduced to that of regular subgroups of Hol(N) which are isomorphic to G, as N ranges over all groups of order |G|, where Hol(−) denotes the holomorph. In this paper, we shall give a description of such subgroups of Hol(N) in terms of bijective crossed homomorphisms G⟶N, and then use it to study two questions related to non-existence of Hopf–Galois structures.