Principal congruence subgroups in the infinite rank case
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910953196486656 |
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| author | Tolstykh, Vladimir A. |
| author_facet | Tolstykh, Vladimir A. |
| contents | We obtain a number of analogues of the classical results of the 1960s on the general linear groups $\mathrm{GL}_n(\mathbf Z)$ and special linear groups $\mathrm{SL}_n(\mathbf Z)$ for the automorphism group $Γ_A=\mathrm{Aut}(A)$ of an infinitely generated free abelian group $A.$ In particular, we obtain a description of normal generators of the group $\mathrm{Aut}(A),$ classify the maximal normal subgroups of the group $\mathrm{Aut}(A),$ describe normal generators of the principal congruence subgroups $Γ_{\!A}(m)$ of the group $\mathrm{Aut}(A),$ and obtain an analogue of Brenner's ladder relation for the group $\mathrm{Aut}(A).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Principal congruence subgroups in the infinite rank case Tolstykh, Vladimir A. Group Theory 20K30 (primary), 20H05, 20F28 (secondary) We obtain a number of analogues of the classical results of the 1960s on the general linear groups $\mathrm{GL}_n(\mathbf Z)$ and special linear groups $\mathrm{SL}_n(\mathbf Z)$ for the automorphism group $Γ_A=\mathrm{Aut}(A)$ of an infinitely generated free abelian group $A.$ In particular, we obtain a description of normal generators of the group $\mathrm{Aut}(A),$ classify the maximal normal subgroups of the group $\mathrm{Aut}(A),$ describe normal generators of the principal congruence subgroups $Γ_{\!A}(m)$ of the group $\mathrm{Aut}(A),$ and obtain an analogue of Brenner's ladder relation for the group $\mathrm{Aut}(A).$ |
| title | Principal congruence subgroups in the infinite rank case |
| topic | Group Theory 20K30 (primary), 20H05, 20F28 (secondary) |
| url | https://arxiv.org/abs/2505.12924 |