The Automorphism Group of the Direct Product of Strongly Related Automata

Abstract

The direct product A X B of two automata A and B has been defined by Rabit~ a~d Scott [1] while the automorphism group of A X B has been investig,~tcd by Fleck [3I The latter showed tha~ the strongly connected automaton A with a transitive abelian auto-m0rphism group G(A) is the direct product of automata if and only if G(A) is isomorphic to the direct product of two groups. The present paper considers somewhat of the reverse problem. If G and H are groups of regular permutations on the finite sets S ~rnd 7' respec~ ~ivcly, there are nonunique strongly connected automata A and B whose automorphism groups are G and H respecLively. To what extent the automorphism group of A )< H is determined by G and H is studied. A sufficient conditiotl that G X H be the group of A X H is produced and it is shown that if G and H are cyclic, there are always automata A ~md B for which G X H is the automorphism group of A X B.