  The axioms of iteration theories, or iteration categories, capture the
equational properties of fixed point operations in several computationally
significant categories. Iteration categories may be axiomatized by the Conway
identities and identities associated with finite automata. We show that in
conjunction with the Conway identities, each identity associated with a finite
automaton implies the identity associated with any input extension of the
automaton. We conclude that the Conway identities and the identities associated
with the members of a subclass $\cQ$ of finite automata is complete for
iteration categories iff for every finite simple group $G$ there is an
automaton $\bQ \in \cQ$ such that $G$ is a quotient of a group in the monoid
$M(\bQ)$ of the automaton $\bQ$. We also prove a stronger result that concerns
identities associated with finite automata with a distinguished initial state.
