### 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 language | English |
---|---|

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

DOIs | |

Publication status | E-pub ahead of print - 25 Feb 2019 |

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**Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras.** / Glasby, S. P.; Ribeiro, Frederico A.M.; Schneider, Csaba.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Duality between p-groups with three characteristic subgroups and semisimple anti-commutative algebras

AU - Glasby, S. P.

AU - Ribeiro, Frederico A.M.

AU - Schneider, Csaba

PY - 2019/2/25

Y1 - 2019/2/25

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, ).

KW - anti-commutative algebras

KW - characteristic subgroups

KW - p-groups

UR - http://www.scopus.com/inward/record.url?scp=85062219627&partnerID=8YFLogxK

U2 - 10.1017/prm.2018.159

DO - 10.1017/prm.2018.159

M3 - Article

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

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

SN - 0308-2105

ER -