Proposition 2

Tag 0AIL, permanent. Cite it as [Bourbaki, Tag 0AIL].

Lie Groups and Lie Algebras, chapter III, § 9. Commutators, centralizers and normalizers in a Lie group

Let G be a topological group and A and B subgroups of G. If A is connected, (A, B) is connected.

For fixed y in B, the set M_y of (x, y) with x ∈ A is connected (for the mapping x ↦ (x, y) of A into G is continuous). e ∈ M_y, and hence the union R of the M_y with y ∈ B is connected. But (A, B) is the subgroup of G generated by R, whence the proposition.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 6, § 9

Cites

Nothing.