As a rule, a finite algebra with" large" automorphism group is functionally complete. Such are, eg, the finite algebras (A; t), where t is the ternary discriminator function on A (H. Werner [23]),(A; d) for IAI-> 3, where d is the dual discriminator on A (E. Fried and AF Pixley [7]). These results were generalized by B. Cs~ k~ iny [4], who proved that almost every nontrivial finite homogeneous algebra (ie, algebra whose automorphism group is the full symmetric group) is functionally complete; up to equivalence, there are only six exceptions. Recently E. Fried, HK Kaiser, and L. Mfirki [6] have given another proof of this result. Their proof works also for infinite algebras, establishing the interpolation property for them. Csfikfiny's theorem was extended to algebras with triply transitive automorphism groups (L. Szab6 and,/~. Szendrei [21], see also HK Kaiser and L. Mgtrki [9]) and later to algebras with doubly transitive automorphism groups (PP P~ lfy, L. Szab6, and i~. Szendrei [11]); the exceptions are the affine spaces over finite fields. Recently J. Demetrovics, L. Hann~ ik and L. R6nyai [5] have shown that an algebra of odd prime order with transitive automorphism group is either functionally complete or its operations are all linear.

It is a simple observation that every finite permutation group is the automorphism group of a functionally complete algebra (Proposition 1). On the other hand, one easily constructs functionally incomplete algebras with given imprimitive automorphism groups (Proposition 2). It seems to us that an imprimitive automorphism group is only a weak constraint on a functionally incomplete algebra, thus most effort is devoted to functionally incomplete algebras with primitive automorphism groups. Their structure is completely described in our