Timeline for Decomposing a compact connected Lie group

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Jun 11, 2020 at 7:39	comment	added	A beginner mathmatician		@fierydemon. I think much simpler argument avoiding covering groups exists. Consider the map $(g,h)\mapsto gh$ from $(Z_G)_0\times G_{ss}$ to $(Z_G)_0G_{ss}.$ The derivative of this map is clearly onto and hence an isomorphism as the corresponding Lie algebras have the same dimension. Thus the map is local diffeomorphism. As the exponential map is a local diffeomorphism and $G$ is connected. The map $(g,h)\mapsto gh$ is surjective.
Jun 8, 2020 at 0:56	comment	added	Ryan Hendricks		You might argue along the lines that $\widetilde{G_{ss}}\times \widetilde{(Z_G)_0}$ is a covering group of $(Z_G)_0 G_{ss}$, that $(Z_G)_0 G_{ss}\subset G$, and $G$ cannot be any bigger than $(Z_G)_0 G_{ss}$ and still have $\widetilde{G_{ss}}\times \widetilde{(Z_G)_0}$ as a covering group
Jun 7, 2020 at 19:37	review	Close votes
Jun 9, 2020 at 18:52
Jun 7, 2020 at 19:16	history	edited	YCor	CC BY-SA 4.0	formatting
Jun 7, 2020 at 19:10	history	asked	A beginner mathmatician	CC BY-SA 4.0