Request: References and/or guidelines to assess the compactness of $D_{\infty h}$

Question

Can the community suggest "good" references for learning how to assess the compactness or lack of thereof in this particular case? Otherwise, I may need guidelines to approach/solve the problem at hand.

Answer 1

It seems that you are dealing with a point group consisting of rotations and reflections, that is, a subgroup of the orthogonal group $O(3)$. The orthogonal group is a closed and bounded subset of the set of square matrices, and thus by the Heine-Borel theorem is compact. It follows that any closed (i.e. contains all its limit points) subgroup of $O(3)$ is also compact (closed subsets of compact sets are compact; this may be found in a Real Analysis text, such as that of Rudin).