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 ${\mathbb F}_p$-algebra $L$ such that the action of ${\rm Aut}(L)$ is irreducible on $L$; we call such an algebra IAC. This paper establishes a duality $G\leftrightarrow L$ 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 ${\rm SL}(2,{\mathbb F})$.