If H is the unique subgroup of a given order in a group G, prove H is characteristic in G.

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- Nov 2nd 2008, 11:01 AMdori1123characteristic group
If H is the unique subgroup of a given order in a group G, prove H is characteristic in G.

- Nov 2nd 2008, 12:06 PMThePerfectHacker
Let $\displaystyle \theta$ be an automorphism of $\displaystyle G$ then $\displaystyle |\theta (H)| = |H|$ - why? But $\displaystyle \phi (H)$ is a subgroup of $\displaystyle H$ because homomorphism images of subgroups are subgroups. And by uniqueness we have $\displaystyle \phi (H) = H$. Therefore, $\displaystyle H$ is invariant under all automorphism i.e. it is a characteristic subgroup.