The Modular Isomorphism Problem -- the alternative perspective on counterexamples
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917745941020672 |
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| author | Bagiński, Czesław Zabielski, Kamil |
| author_facet | Bagiński, Czesław Zabielski, Kamil |
| contents | As a result of impressive research arXiv:2106.07231, D. Garc\'ıa-Lucas, Á. del Río and L. Margolis defined an infinite series of non-isomorphic $2$-groups $G$ and $H$, whose group algebras $\mathbb{F}G$ and $\mathbb{F}H$ over the field $\mathbb{F}=\mathbb{F}_2$ are isomorphic, solving negatively the long-standing Modular Isomorphism Problem (MIP). In this note we give a different perspective on their examples and show that they are special cases of a more general construction. We also show that this type of construction for $p>2$ does not provide a similar counterexample to the MIP. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01810 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Modular Isomorphism Problem -- the alternative perspective on counterexamples Bagiński, Czesław Zabielski, Kamil Group Theory Rings and Algebras 16S34, 16U60, 20C05, 20D15 As a result of impressive research arXiv:2106.07231, D. Garc\'ıa-Lucas, Á. del Río and L. Margolis defined an infinite series of non-isomorphic $2$-groups $G$ and $H$, whose group algebras $\mathbb{F}G$ and $\mathbb{F}H$ over the field $\mathbb{F}=\mathbb{F}_2$ are isomorphic, solving negatively the long-standing Modular Isomorphism Problem (MIP). In this note we give a different perspective on their examples and show that they are special cases of a more general construction. We also show that this type of construction for $p>2$ does not provide a similar counterexample to the MIP. |
| title | The Modular Isomorphism Problem -- the alternative perspective on counterexamples |
| topic | Group Theory Rings and Algebras 16S34, 16U60, 20C05, 20D15 |
| url | https://arxiv.org/abs/2406.01810 |