Lemma 4

Tag 09H2, permanent. Cite it as [Bourbaki, Tag 09H2].

Integration, VII, p. 59, § 3. Applications and examples

Let G be a connected topological group, D a totally disconnected normal subgroup of G. Then D is contained in the center of G.

For, let d ∈ D. The image of G under the continuous mapping x ↦ xdx^{-1} is a connected subset of D, hence reduces to {d}, which proves that xd = dx for all x ∈ G.

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

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