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

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Suppose G is a periodic nilpotent group. We know that this means that G is a restricted external direct product of nilpotent p-groups. The automorphism group of G is isomorphic to the (unrestricted) external direct product of the automorphism groups of each of the nilpotent p-groups.

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