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

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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.

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