I don't understand what's the motivation to define group actions, from what I can understand we associate a group with a set, but that doesn't tell me much. Can I have an example that shows how useful group action is? Can the set be a lie group ?
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One further example is the proof of the Sylow theorems using group actions. It makes the proof much easier and much better to understand. – Dietrich Burde Jul 05 '21 at 11:24
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is there a connection between lie groups and group actions ? – 領域展開 Jul 05 '21 at 11:29
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Yes, there is. A Lie group is a group, after all. It can act on manifolds, for example. See Lie group action. – Dietrich Burde Jul 05 '21 at 11:30
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Good that alone it's important, where can I find some examples about group actions in lie groups and manifolds ?, what domain studies that ? – 領域展開 Jul 05 '21 at 11:34
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That's differential geometry and Lie theory. Google for "Lie group actions". – Dietrich Burde Jul 05 '21 at 11:36
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Ok will do, thanks a lot for helping. – 領域展開 Jul 05 '21 at 11:37