Quasi-inverse endomorphisms

TL;DR

This paper explores the relationship between fixed-point-free endomorphisms with abelian images and their inverses under a specific operation, providing a method to construct all such endomorphisms in the context of Hopf Galois theory.

Contribution

It establishes a characterization of fixed-point-free endomorphisms with abelian images via their inverses in a near-ring, and offers a construction method for these endomorphisms.

Findings

A fixed-point-free endomorphism has an abelian image if and only if it has an inverse under the circle operation.

Provides an explicit recipe for constructing all such endomorphisms.

Connects the theory to Hopf Galois structures and near-ring algebra.

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 holomorph 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.