Fixed-point free endomorphisms and Hopf Galois structures

Abstract

Let L | K L|K be a Galois extension of fields with finite Galois group G G . Greither and Pareigis showed that there is a bijection between Hopf Galois structures on L | K L|K and regular subgroups of P e r m ( G ) Perm(G) normalized by G G , and Byott translated the problem into that of finding equivalence classes of embeddings of G G in the holomorph of groups N N of the same cardinality as G G . In 2007 we showed, using Byott’s translation, that fixed point free endomorphisms of G G yield Hopf Galois structures on L | K L|K . Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the Greither-Pareigis approach and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are “twistings” of the Hopf Galois structure by H λ H_{\lambda } , the K K -Hopf algebra that arises from the left regular representation of G G in P e r m ( G ) Perm(G) . The paper concludes with various old and new examples of abelian fixed point free endomorphisms.

Quasi-inverse endomorphisms

“…In this paper we study the fixed-point-free endomorphisms of a finite group that have a quasi-inverse. We will show that for a fixed-point-free endomorphism of a finite group the properties of being abelian, and that of having a quasiinverse, are equivalent (Theorem 3.4, which extends [Chi12, Remark 10], and shows that the condition of having a quasi-inverse is not restrictive in the context of [Chi12]). In Sections 4 and 7 we are able to give reasonably explicit recipes for constructing the groups that have such an endomorphism, and for determining all of their endomorphisms with this property.…”

The Structure of Hopf Algebras Acting on Dihedral Extensions

“…Some specific cases-most notably the cases where Gal(L/K) is cyclic or elementary abelian-have been investigated by the authors in [17]. We note that L. Childs [10,Theorem 5] has shown that abelian fixed-point free endomorphisms of Gal(L/K) determine Hopf Galois structures on L/K whose Hopf algebras are isomorphic to H λ (see Example 2.4 for the definition of H λ ). Childs applies his result to the cases where Gal(L/K) is the symmetric group S n , n ≥ 5, and the dihedral group of order 4n, n ≥ 2.…”

Non-isomorphic Hopf Galois structures with isomorphic underlying Hopf algebras

“…Let us note that it may be checked that all separable Hopf Galois extensions of degree up to seven are almost classically Galois (see [6]) and that all separable extensions of degree 8 whose Galois closure has degree 16 are also almost classically Galois (using Magma). For a degree 8 extension whose Galois closure has degree 24, the Galois group G of its Galois closure is isomorphic either to the symmetric group S 4 , the special linear group SL (2,3) or the direct product A 4 × C 2 of the alternating group A 4 and the cyclic group C 2 of order 2. One may check that all degree 8 extensions in the last two cases are also almost classically Galois.…”

“…In the last section, we observe that the image of the Galois correspondence of a Hopf Galois structure does not determine the isomorphism class of the underlying Hopf algebra. Examples of non-isomorphic Hopf Galois structures with isomorphic underlying Hopf algebras have previously be given in [4] and [3] all of them for Galois extensions K/k and the Hopf algebra H 1 that gives the standard non-classical Hopf Galois structure, obtained by taking N = λ(G). Let us note that these examples are obtained by counting the number of different Hopf Galois structures on K/k given by H 1 whereas in this paper we are giving explicitly both the Hopf algebra and the Hopf Galois structures.…”

“…To make the Hopf actions μ 3 and μ 4 of H 3 and H 4 explicit, we first compute the preimage of 1 G under the elements of N 3 and N 4 .…”