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Let $G$ be a group.
Let $H$ be a subgroup such that:
- $\forall \phi \in \Aut G: \phi \sqbrk H = H$
where $\Aut G$ is the automorphism group of $G$.
Then $H$ is characteristic (in $G$), or a characteristic subgroup of $G$.
- Results about characteristic subgroups can be found here.