I'm working on a presentation on simple Lie groups and would like to show by example that the simple Lie groups are connected, but I'm not really sure how to do this. I'd also like to show that one of $SU(n),SO(n)$, or $Sp(n)$ is compact as well. Any suggestions on how I should go about doing this?
2026-04-01 12:35:57.1775046957
Showing a simple Lie group is connected and compact.
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The connected component of the identity element of a topological group is a normal subgroup. If the group is simple (in the sense of group theory), there are very few candidates for this subgroup.
To show compactness, simply show that those groups are homeomorphic to specific closed and bounded subsets of $\mathbb R^m$ for appropriate $m$.