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In the finite group theory, if we ignore the group structure, we can tell that the group can be written as the direct product of the subgroup and the representative elements of the cosets. However, even if we ignore the group structure, the lie group can't always be written as the direct product of the subgroup and the coset space. For example, $SO(3)\neq SO(2)\times S^2$. So I'm wondering whether the coset space is equivalent to the coset, if not, is it possible to derive the coset of a lie group?

金广羊
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  • If you ignore the group and manifold structure, (i.e., just look at set theoretic bijections), then ,$SO(3)=SO(2)\times S^2$ is true in the same sense it is for finite groups. – Jason DeVito - on hiatus May 30 '21 at 12:29
  • I don't think when discussing the coset we can abandon the manifold structure, since the group multiplication is defined on the product manifold... – 金广羊 Jun 01 '21 at 15:24
  • But my point is that if you abandon the group structure on a group, all that is left is a set. The analogous idea for Lie groups is to abandon both the group structure and manifold structure. I guess I just don't understand what your question is. – Jason DeVito - on hiatus Jun 01 '21 at 16:41

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