If a subgroup is unique of a given order then it is characteristic

If H is the unique subgroup of order |H| in G, prove that H is characteristic.


Let \varphi \in \mathsf{Aut}(G) and let H \leq G; suppose H is the unique subgroup of order |H|. Now \varphi|_H is an injection of H onto a subgroup of order |H| in G; hence \varphi[H] = H. Thus H is characteristic.

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