Fuchs' problem for endomorphisms of nonabelian groups

TL;DR

This paper explores which nonabelian groups can be fully realized as groups of units in rings, extending previous work on abelian groups by constructing infinite families of such groups through semidirect products.

Contribution

It provides a comprehensive analysis of fully realizable nonabelian groups, including symmetric, dihedral, quaternion, and other specific families, and introduces new constructions using iterated semidirect products.

Findings

Identifies several families of nonabelian groups that are fully realizable.

Constructs three infinite families of fully realizable nonabelian groups.

Extends the concept of fully realizable groups beyond abelian cases.

Abstract

In 1960, L\'{a}szl\'{o} Fuchs posed the problem of determining which groups $G$ are realizable as the group of units in some ring $R$. In \cite{chebolu2022fuchs}, we investigated the following variant of Fuchs' problem, for abelian groups: which groups $G$ are realized by a ring $R$ where every group endomorphism of $G$ is induced by a ring endomorphism of $R$? Such groups are called fully realizable. In this paper, we answer the aforementioned question for several families of nonabelian groups: symmetric, dihedral, quaternion, alternating, and simple groups; almost cyclic $p$-groups; and groups whose Sylow $2$-subgroup is either cyclic or normal and abelian. We construct three infinite families of fully realizable nonabelian groups using iterated semidirect products.