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

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 F 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, F).