The multiple holomorph of a finitely generated abelian group

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?