Fuchs' problem for endomorphisms of nonabelian groups

In 1960, L & aacute;szl & oacute; Fuchs posed the problem of determining which groups G are realizable as the group of units in some ring R. In [4], 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 pgroups; 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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.