Let $G$ be a Lie group, $H$ a subgroup of $G$, which is an immersed submanifold of $G$.

Kirillov, in his book "Introduction to Lie Groups and Lie Algebras" claims* it's easy to see $H$ will be a Lie group.

I do not see why the multiplication in $H$ has to be smooth.

Denote by $m:G \times G \to G$ the multiplication in $G$.

$H$ is an immersed submanifold in $G \Rightarrow$ $H \times H$ is an immersed submanifold in $G \times G \Rightarrow$

$m|_{H \times H}:H \times H \to G$ is smooth**.

Now, we would like to say $m|_{H \times H}$ remains smooth after restricting the codomain to $H$. However, restriction of the codomain preserves smoothness if and only if continuity is preserved***.

$H$ can contain open sets which are not in the subspace topology on it, so this restriction does not have to be continuous.

Question: Is it true that $H$ is always a Lie group?

*See page 17 (Definition 2.14). In fact Kirillov defines a Lie subgroup as an immersed submanifold which is also a subgroup.

**See theorem 5.27 in Lee's book Introduction to smooth manifolds

***See theorem 5.29 in Lee's book

  • $\begingroup$ See Definition $7.3.3$ and Theorems $7.3.4$ and $7.3.5$ here. $\endgroup$ – Dietrich Burde Nov 9 '15 at 20:11

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