Identification of non-isomorphic 2-groups with dihedral central quotient and isomorphic modular group algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916894367285248 |
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| author | Margolis, Leo Sakurai, Taro |
| author_facet | Margolis, Leo Sakurai, Taro |
| contents | The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by García-Lucas, Margolis and del Río [J. Reine Angew. Math. 783 (2022), pp. 269-274]. We embed these negative solutions in the class of two-generated finite $2$-groups with dihedral central quotient, and solve the original question for all groups within this class. As a result, we discover new negative solutions and simple algebra isomorphisms. At the same time, the positive solutions for most of the groups in this class give some insights what makes the negative solutions special. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Identification of non-isomorphic 2-groups with dihedral central quotient and isomorphic modular group algebras Margolis, Leo Sakurai, Taro Rings and Algebras Group Theory 20C05 (Primary) 16S34, 20D15, 16U60 (Secondary) The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by García-Lucas, Margolis and del Río [J. Reine Angew. Math. 783 (2022), pp. 269-274]. We embed these negative solutions in the class of two-generated finite $2$-groups with dihedral central quotient, and solve the original question for all groups within this class. As a result, we discover new negative solutions and simple algebra isomorphisms. At the same time, the positive solutions for most of the groups in this class give some insights what makes the negative solutions special. |
| title | Identification of non-isomorphic 2-groups with dihedral central quotient and isomorphic modular group algebras |
| topic | Rings and Algebras Group Theory 20C05 (Primary) 16S34, 20D15, 16U60 (Secondary) |
| url | https://arxiv.org/abs/2405.08075 |