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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. 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