Automorphism group of finite nilpotent group is direct product of automorphism groups of Sylow subgroups
Suppose is a finite nilpotent group. Suppose the prime factors of are and the corresponding Sylow subgroups are respectively . Then, the automorphism group is isomorphic to the external direct product of the automorphism groups . Explicitly, in the corresponding internal direct product, the direct factor corresponding to is the subgroup of comprising those automorphism that fix all the for .