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

S. P. Glasby, Frederico A.M. Ribeiro, Csaba Schneider

Research output: Contribution to journalArticle

Abstract

Let p be an odd prime and let G be a non-abelian finite p-group of exponent p 2 with three distinct characteristic subgroups, namely 1, G p and G. The quotient group G/G p 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 â†" 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, ).

Original languageEnglish
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
DOIs
Publication statusE-pub ahead of print - 25 Feb 2019

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Anti-commutative
Commutative Algebra
P-groups
Semisimple
Duality
Subgroup
Algebra
Quotient group
Finite P-group
Odd
Classify
Exponent
Distinct
Module

Cite this

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abstract = "Let p be an odd prime and let G be a non-abelian finite p-group of exponent p 2 with three distinct characteristic subgroups, namely 1, G p and G. The quotient group G/G p 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 {\^a}†{"} 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, ).",
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AU - Ribeiro, Frederico A.M.

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N2 - Let p be an odd prime and let G be a non-abelian finite p-group of exponent p 2 with three distinct characteristic subgroups, namely 1, G p and G. The quotient group G/G p 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 â†" 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, ).

AB - Let p be an odd prime and let G be a non-abelian finite p-group of exponent p 2 with three distinct characteristic subgroups, namely 1, G p and G. The quotient group G/G p 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 â†" 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, ).

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JF - PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH: SECTION A MATHEMATICS

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