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

Part of: General nonassociative rings Representation theory of groups Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  25 February 2019

S. P. Glasby ,
Frederico A. M. Ribeiro  and
Csaba Schneider
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 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, 𝔽).

Keywords

characteristic subgroupsp-groupsanti-commutative algebras

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups 20C20: Modular representations and characters
Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups 20F28: Automorphism groups of groups 17A30: Algebras satisfying other identities 17A36: Automorphisms, derivations, other operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1827 - 1852
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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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