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

Published online by Cambridge University Press:  25 February 2019

S. P. Glasby
Affiliation:
Centre for Mathematics of Symmetry and Computation, University of Western Australia, 35 Stirling Highway, Perth6009, Australia (Stephen.Glasby@uwa.edu.au)
Frederico A. M. Ribeiro
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, Brazil (fred321@gmail.com; csaba@mat.ufmg.br)
Csaba Schneider
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, Brazil (fred321@gmail.com; csaba@mat.ufmg.br)
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Abstract

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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 𝔽p-algebra L such that the action of Aut (L) is irreducible on L; we call such an algebra IAC. This paper establishes a duality GL 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, 𝔽).

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

Footnotes

*

Current address: Departamento de Matemática, Centro Federal de Educação, Técnológica de Minas Gerais, CEFET-MG, Av. Amazonas 7675, Belo, Horizonte, MG, Brasil.

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