The properties of endomorphisms and automorphisms of a finite, deterministic automatonA related to the smallest input-independent partition on the set of internal states ofA are investigated. The setH d of all thed-endomorphisms ofA defined here, as well as the setG d of all thed-automorphisms ofA, are studied in detail. It is proved thatH d forms a polyadic semigroup, whileG d forms a polyadic group. Connections betweenG d and the groupG(A) of all the automorphisms ofA are examined. The upper bound for the cardinality ofG d is given. Finally, by means of the theory ofd-automorphisms, some problems of the theory of strictly periodic automata are solved; in the first place, the necessary and sufficient condition for the reducibility of an arbitrary strictly periodic automation is given.