Compact subgroups of \(\text{GL}_n(\mathbb C)\) (Q876745)

The authors prove the following theorem: Let \(G\subset \text{GL}_n(\mathbb C)\) be a closed subgroup such that each element of \(G\) is semisimple with all eigenvalues having absolute value 1. Then \(G\) is conjugated to a subgroup of the unitary group and hence, in particular, compact. In the connected case, the proof is based on Lie algebra techniques, while the general case then follows with a result of Schur about torsion subgroups.