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arXiv · 2405.08075

Identification of non-isomorphic 2-groups with dihedral central quotient and isomorphic modular group algebras

Leo Margolis·Taro Sakurai
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Abstract

The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by Garc\'ia-Lucas, Margolis and del R\'io [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.

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Leo Margolis, Taro Sakurai. 2024-05-13. Identification of non-isomorphic 2-groups with dihedral central quotient and isomorphic modular group algebras. https://doi.org/10.4171/rmi%2F1531

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