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Let $G$ be a Lie group and $H$ a Lie subgroup. Then $H$ contains the connected component of the identity $G^0$ if and only if $G/H$ is a discrete manifold.

One direction is immediate since $G/G^0$ is discrete. Now if $G/H$ is discrete then $H$ is open and obviously contains the identity so it contains an open set $H\cap G^0$. Not sure how to show that it contains all of $G^0$ unless I have connectedness of $H$ though.

Dian_Horton92
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    An open neighborhood of the identity generates the whole connected component of the identity, which you show using the exponential map. – Mikhail Katz Jul 30 '23 at 16:18
  • @MikhailKatz The answer below is more general and solves the problem but this comment is actually more direct. – Dian_Horton92 Jul 31 '23 at 00:09

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$H$ is in fact clopen. It is a more general fact that a clopen subset of any topological space is a union of connected components (of that topological space): Why are clopen sets a union of connected components?

In particular this implication holds for any topological group. But of course the other one does not.

freakish
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  • Perhaps important to mention here are quasi-components of a topological space. A quasi-component of point $x$ is the intersection of all clopen subsets containing $x$. The component containing $x$ is always contained in the quasi-component. – Jakobian Jul 30 '23 at 16:33