The automorphism group of a product of graphs

Abstract

In a recent paper we showed that every connected graph can be written as a weak cartesian product of a family of indecomposable rooted graphs and that this decomposition is unique to within isomorphisms. Using this unique prime factorization theorem we prove that if a graph X can be writteni as a product of connected rooted graphs, which are pairwise relatively prime, then the automorphism group of X is isomorphic to the restricted direct product of the automorphism groups of the factors with prescribed subgroups the isotropy groups of the factors at the roots. This is a generalization of Sabidussi's theorem for cartesian multiplication. Sabidussi [4] proved that if Xi, , X. are connected graphs of finite type which are pairwise relatively prime with respect to cartesian multiplication, then the automorphism group of the product is isomorphic to the direct product of the automorphism groups of the factors. The proof of this result uses the unique prime factorization theorem for cartesian multiplication. In [3] it was proved that, for connected graphs, unique prime factorization holds for weak cartesian multiplication. The purpose of this note is to generalize Sabidussi's Theorem to the weak cartesian product of a family of rooted graphs. Let (Ga)aeA be a family of groups and for each a(A let Ha be a subgroup of Ga. The restricted direct product of the family of groups (Ga)aEA with prescribed subgroups Ha is defined by II (Ga, Ha) {gE H Ga I pra gHa for all but finitely many aA}. aE-A aE-A This product was introduced by Vilenkin [6] and can be found in Kuros [2 ]. For a subgraph Y of a graph X we use the notation of [5 ] and denote the subgroup of the automorphism group of X that leaves Y invariant by G(X; Y), that is, G(X; Y)= {IkEG(X)|IkY= Y}I where G(X) is the automorphism group of X. Hence G(Xa; xa) denotes the isotropy group of Xa at xa. Our main result is the following: THEOREM. Let (Xa, Xa)aGA be a family of connected rooted graphs which are pairwise relatively prime. Then G(TjaEA (Xa, xa)) is isomorphic to Received by the editors July 30, 1969. A MS Subject Classifications. Primary 0562; Secondary 2022, 2052.