Let $G$ be a non-discrete Lie group. Can the connected component of its identity ($G_e$) be presented as the union of its Lie connected subgroups of dimension 1: $$G_e = \bigcup\{ \text{Lie connected subgroups of } G \text{ of dimension } 1 \}?$$
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4No. 1 dimensional connected subgroups can be shown to be 1-parameter subgroups. So they must be in the image of $\exp$ and that is not surjective in general. – Callum Nov 07 '23 at 13:40
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@Callum What about those isomorphic to $\mathbb{S}^1$? – Lazarus Frost Nov 07 '23 at 22:42
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What about them? They are still 1parameter subgroups. – Callum Nov 08 '23 at 07:17
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@Callum Oh, I didn't know that they are also 1-parameter groups. – Lazarus Frost Nov 08 '23 at 21:41