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Let $p$ be an odd prime and let $G$ be a non-abelian finite $p$-group ofexponent $p^2$ with three distinct characteristic subgroups, namely $1$, $G^p$,and $G$. The quotient group $G/G^p$ gives rise to an anti-commutative ${\mathbbF}_p$-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 $G\leftrightarrowL$ between such groups and such IAC algebras. We prove that IAC algebras aresemisimple and we classify the simple IAC algebras of dimension at most 4 overcertain fields. We also give other examples of simple IAC algebras, including afamily related to the $m$-th symmetric power of the natural module of ${\rmSL}(2,{\mathbb F})$.

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