Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras

Let p be an odd prime and let G be a non-abelian finite p-group of exponent p2 with three distinct characteristic subgroups, namely 1, Gp and G. The quotient group G/Gp gives rise to an anti-commutative p-algebra L such that the action of Aut (L) is irreducible on L; we call such an algebra IAC. Thi...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2020-08, Vol.150 (4), p.1827-1852
Main Authors: Glasby, S. P., Ribeiro, Frederico A. M., Schneider, Csaba
Format: Article
Language:English
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Summary:Let p be an odd prime and let G be a non-abelian finite p-group of exponent p2 with three distinct characteristic subgroups, namely 1, Gp and G. The quotient group G/Gp gives rise to an anti-commutative p-algebra L such that the action of Aut (L) is irreducible on L; we call such an algebra IAC. This paper establishes a duality G ↔ L between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the m-th symmetric power of the natural module of SL(2, ).
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2018.159