Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups

Abstract

For G any finite group, the left and right regular representations λ, respectively ρ of G into Perm ( G ) map G into InHol ( G ) = ρ ( G ) ⋅ Inn ( G ) . We determine regular embeddings of G into InHol ( G ) modulo equivalence by conjugation in Hol ( G ) by automorphisms of G, for groups G that are semidirect products G = Z h ⋊ Z k of cyclic groups and have trivial centers. If h is the power of an odd prime p, then the number of equivalence classes of regular embeddings of G into InHol ( G ) is equal to twice the number of fixed-point free endomorphisms of G, and we determine that number. Each equivalence class of regular embeddings determines a Hopf Galois structure on a Galois extension of fields L / K with Galois group G. We show that if H 1 is the Hopf algebra that gives the standard non-classical Hopf Galois structure on L / K , then H 1 gives a different Hopf Galois structure on L / K for each fixed-point free endomorphism of G.