Fuchs' problem for endomorphisms of nonabelian groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911989890023424 |
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| author | Chebolu, Sunil K. Lockridge, Keir |
| author_facet | Chebolu, Sunil K. Lockridge, Keir |
| contents | In 1960, László 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fuchs' problem for endomorphisms of nonabelian groups Chebolu, Sunil K. Lockridge, Keir Group Theory Rings and Algebras In 1960, László 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. |
| title | Fuchs' problem for endomorphisms of nonabelian groups |
| topic | Group Theory Rings and Algebras |
| url | https://arxiv.org/abs/2408.08195 |