April 16, 2013
Abstract. Greither and Pareigis have established a connection between Hopf Galois structures on a Galois extension L / K with Galois group G , and the regular subgroups of the group of permutations on G , which are normalized by G . Byott has rephrased this connection in terms of certain equivalence classes of injective morphisms of G into the holomorphs of the groups N with the same cardinality of G . Childs and Corradino have used this theory to construct such Hopf Galois structures, starting from fixed-point-free endomorphisms of G that have abelian images. In this paper we show that a fixed-point-free endomorphism has an abelian image if and only if there is another endomorphism that is its inverse with respect to the circle operation in the near-ring of maps on G , and give a fairly explicit recipe for constructing all such endomorphisms.