The multiple holomorph of a finitely generated abelian group

TL;DR

This paper provides an alternative method to classify regular subgroups of the symmetric group related to a finitely generated abelian group, using commutative ring structures, and explores automorphisms preserving these structures.

Contribution

It introduces a new approach based on commutative ring structures to analyze the holomorphs of finitely generated abelian groups, extending Mills' results.

Findings

Characterization of regular subgroups via commutative rings

Identification of ring structures where automorphisms are preserved

Solution to automorphism-preserving ring structure classification

Abstract

W.H.~Mills has determined, for a finitely generated abelian group $G$, the regular subgroups $N \cong G$ of $S(G)$, the group of permutations on the set $G$, which have the same holomorph of $G$, that is, such that $N_{S(G)}(N) = N_{S(G)}(\rho(G))$, where $\rho$ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of $N_{S(G)}(\rho(G))$ in terms of commutative ring structures on $G$. We are led to solve, for the case of a finitely generated abelian group $G$, the following problem: given an abelian group $(G, +)$, what are the commutative ring structures $(G, +, \cdot)$ such that all automorphism of $G$ as a group are also automorphisms of $G$ as a ring?