Abstract. An algebra is characteristically simple if it has no nontrivial congruence relations preserved by all automorphisms. We investigate and classify the finite characteristically simple algebras. From the main results, among others, we derive an explicit description of the finite characteristically simple algebras with transitive automorphism groups.

AMS Subject Classification (1991): 08A40

Received September 13, 1995, in final form January 6, 1996. (Registered under 6102/2009.)