On the Modular Isomorphism Problem for 2-generated groups with cyclic derived subgroup
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| Format: | Preprint |
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2023
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| author | García-Lucas, Diego del Río, Ángel |
| author_facet | García-Lucas, Diego del Río, Ángel |
| contents | We continue the analysis of the Modular Isomorphism Problem for $2$-generated $p$-groups with cyclic derived subgroup, $p>2$, started in [D. García-Lucas, Á. del Río, and M. Stanojkovski. On group invariants determined by modular group algebras: even versus odd characteristic. Algebr. Represent. Theory. https://doi.org/10.1007/s10468-022-10182-x, 2022]. We show that if $G$ belongs to this class of groups, then the isomorphism type of the quotients $G/(G')^{p^3}$ and $G/γ_3(G)^p$ are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification \cite{OsnelDiegoAngel}. We also show that for groups in this class of order at most $p^{11}$, the Modular Isomorphism Problem has positive answer. Finally, we describe some families of groups of order $p^{12}$ whose group algebras over the field with $p$ elements cannot be distinguished with the techniques available to us. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_02627 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Modular Isomorphism Problem for 2-generated groups with cyclic derived subgroup García-Lucas, Diego del Río, Ángel Group Theory 20D15 We continue the analysis of the Modular Isomorphism Problem for $2$-generated $p$-groups with cyclic derived subgroup, $p>2$, started in [D. García-Lucas, Á. del Río, and M. Stanojkovski. On group invariants determined by modular group algebras: even versus odd characteristic. Algebr. Represent. Theory. https://doi.org/10.1007/s10468-022-10182-x, 2022]. We show that if $G$ belongs to this class of groups, then the isomorphism type of the quotients $G/(G')^{p^3}$ and $G/γ_3(G)^p$ are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification \cite{OsnelDiegoAngel}. We also show that for groups in this class of order at most $p^{11}$, the Modular Isomorphism Problem has positive answer. Finally, we describe some families of groups of order $p^{12}$ whose group algebras over the field with $p$ elements cannot be distinguished with the techniques available to us. |
| title | On the Modular Isomorphism Problem for 2-generated groups with cyclic derived subgroup |
| topic | Group Theory 20D15 |
| url | https://arxiv.org/abs/2310.02627 |