It has been shown that the axioms of iteration theories capture the equational properties of iteration in several different models related to computer science. Iteration theories are axiomatizable by the Conway identities and the group-identities corresponding to the finite (simple) groups. In this paper we provide a complete analysis of these identities by giving a concrete description of the free theories in the variety axiomatized by the Conway identities and any given subcollection of the group-identities. It follows that when the group-identities are effectively given, the equational theory of the variety is decidable.
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