Show that for any subgroup H < G where G is a compact Lie-group, the conjugacy groups $\{gHg^{-1} |\: g \in G\}$ form compact embedded submanifolds.
My ideas:
I am wondering whether "H is closed" is needed as an assumption. Because then we would have since the conjugation-action is a diffeomorphism, that any conjugacy-group is also closed. And as a subspace of G therefore compact.
If H would be closed, then since the conjugation-action $x \mapsto g \cdot x \cdot g^{-1}$ is a homeomorphism, its image is also closed in G. It is easy to verify, that $\{ghg^-1 | h \in H\}$ satisfies the group axioms for any $g \in G$. And by the closed-subgroup theorem, it would therefore be an embedded submanifold.
But I cannot find a way to solve this without H closed. If anyone has an idea whether that would work that would be really helpful.
Thanks in advance !