# Automorphism group of finite nilpotent group is direct product of automorphism groups of Sylow subgroups

Suppose $G$ is a finite nilpotent group. Suppose the prime factors of $G$ are $p_1,p_2,\dots,p_r$ and the corresponding Sylow subgroups are respectively $P_1,P_2,\dots,P_r$. Then, the automorphism group $\operatorname{Aut}(G)$ is isomorphic to the external direct product of the automorphism groups $\operatorname{Aut}(P_i)$. Explicitly, in the corresponding internal direct product, the direct factor corresponding to $\operatorname{Aut}(P_i)$ is the subgroup of $\operatorname{Aut}(G)$ comprising those automorphism that fix all the $P_j$ for $j \ne i$.