Fuchs' problem for linear groups
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866908836393123840 |
|---|---|
| author | Lockridge, Keir Terkel, Jacinda |
| author_facet | Lockridge, Keir Terkel, Jacinda |
| contents | Which groups can occur as the group of units in a ring? Such groups are called realizable. Though the realizable members of several classes of groups have been determined (e.g., cyclic, odd order, alternating, symmetric, finite simple, indecomposable abelian, and dihedral), the question remains open. The general linear groups are realizable by definition: they are the units in the corresponding matrix rings. In this paper, we study the realizability of two closely related linear groups, the special linear groups and the affine general linear groups. We determine which special linear groups of degree 2 over a finite field are realizable by a finite ring, and we determine which affine general linear groups of degree 1 over a cyclic group are realizable by a finite ring. We also give partial results for certain linear groups of other degrees and for rings of characteristic zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fuchs' problem for linear groups Lockridge, Keir Terkel, Jacinda Group Theory Rings and Algebras 16U60, 20G40, 16S34 Which groups can occur as the group of units in a ring? Such groups are called realizable. Though the realizable members of several classes of groups have been determined (e.g., cyclic, odd order, alternating, symmetric, finite simple, indecomposable abelian, and dihedral), the question remains open. The general linear groups are realizable by definition: they are the units in the corresponding matrix rings. In this paper, we study the realizability of two closely related linear groups, the special linear groups and the affine general linear groups. We determine which special linear groups of degree 2 over a finite field are realizable by a finite ring, and we determine which affine general linear groups of degree 1 over a cyclic group are realizable by a finite ring. We also give partial results for certain linear groups of other degrees and for rings of characteristic zero. |
| title | Fuchs' problem for linear groups |
| topic | Group Theory Rings and Algebras 16U60, 20G40, 16S34 |
| url | https://arxiv.org/abs/2401.11583 |