To amplify Ben's answer, I'd point to an earlier post that has lots more detail: here. The subject of compact groups is old and well-studied, so there are many references to choose from, even Wikipedia perhaps. Anyway, it's good to browse older Lie group entries on MO first.

PS: This supplementary "answer" is really a suggestion that the question is too close to the earlier post I cited to qualify as a fresh question. Textbook material of this kind calls mainly for references rather than discussion.

To amplify Ben's answer, I'd point to an earlier post that has lots more detail: here. The subject of compact groups is old and well-studied, so there are many references to choose from, even Wikipedia perhaps. Anyway, it's good to browse older Lie group entries on MO first.

PS: This supplementary "answer" is really a suggestion that the question is too close to the earlier post I cited to qualify as a fresh question. Textbook material of this kind calls mainly for references rather than discussion.

To amplify Ben's answer, I'd point to an earlier post that has lots more detail: here. The subject of compact groups is old and well-studied, so there are many references to choose from, even Wikipedia perhaps. Anyway, it's good to browse older Lie group entries on MO first.

To amplify Ben's answer, I'd point to an earlier post that has lots more detail: here. The subject of compact groups is old and well-studied, so there are many references to choose from, even Wikipedia perhaps. Anyway, it's good to browse older Lie group entries on MO first.

PS: This supplementary "answer" is really a suggestion that the question is too close to the earlier post I cited to qualify as a fresh question. Textbook material of this kind calls mainly for references rather than discussion.