Point group of a disjoint union of graphs Let $G$ be a graph. $\Gamma(G)$ is the point group og $G$, i.e. the automorphism group of $G$. Suppose $$G \cong nH $$ i.e. the disjoint union of $n$ graphs isomorphic to $H$. Then what is $\Gamma(G)$ ?

Let $H$ be a connected graph and let $G$ consist of $n$ copies of $H$. The point group $\Gamma(G)$ contains elements that map each copy of $H$ to itself. Thus, the $n$-fold direct product $\Gamma(H)^n$ is a subgroup of $\Gamma(G)$. Also, the different copies of $H$ can be permuted amongst themselves, giving another $n!$ automorphisms $S_n$. These two subgroups together generate a group called the wreath product $\Gamma(H) Wr S_n = \Gamma(H)^n \rtimes S_n$. Thus the graph $nH$ has $n! |\Gamma(H)|^n$ automorphisms.