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Compact subgroups of \(\text{GL}_n(\mathbb C)\). (English) Zbl 1115.22004

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.

MSC:

22C05 Compact groups
22E10 General properties and structure of complex Lie groups
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References:

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