Intersection of a locally finite subgroup and a compact subgroup of a linear group.

Question

The following question(s) is due to my complete ignorance on the subject. I apology in advance if they will be silly questions for the experts.

Let $K$ be a topological field and consider the topological group $G = \mathrm{GL}_n(K)$. Let $N$ be the unique maximal normal locally finite subgroup of $G$ and let $H$ be a compact subgroup of $G$.

• is $N \cap H$ always non-empty?
• if the intersection is non-empty, then is $N \cap H$ open in H?

Answer 1

Well, $N\cap H$ is certainly nonempty, since it contains the identity element. But it does not need to be open in $H$. For instance let $K=\mathbb{C}$ and $n=1$. Then $N$ is the group of roots of unity, but $H$ could be the entire circle group $U(1)$, which contains $N$ as a dense subset with empty interior.