can we say something about the automorphism group of a graph $G$ that has the property: $ G \cong A + B $ , if we know the automorphism groups of $A$ and $B$ respectively. The $+$ is the union $ \cup$ of the $A$ and $B$ with the only addition that $ V(A) \cap V(B)= \emptyset$.
Thank you in advance, any view on this would be helpful!
Assuming A is not isomorphic to B, the automorphism group is the direct product of the groups of A and B. The direct product can be thought of as the set of elements (a,b) for a in the group for A and b in the group for B. Then (a,c) times (b,d) is (ab,cd), because the automorphism of B does not affect that of A and vice versa.