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

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, ).