CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Compact connected abelian subgroups lie in maximal tori

Statement

Assume the Axiom of Choice. Every compact connected abelian subgroup of a compact Lie group is contained in a maximal torus.

Facts & Assumptions

Given: Assume the Axiom of Choice, a compact Lie group $G$ , and a subgroup $A \leq G$ that is compact, connected and abelian.

[A1]

The Axiom of Choice is The Axiom of Choice; it supplies the countable choice assumed by the closed-subgroup theorem in [L2] and supplies the stated hypothesis of [L3], whose current maximum-dimension proof needs no choice.

[L1]

A compact subset of a Hausdorff space is closed (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones).

[L2]

A closed subgroup of a finite-dimensional real Lie group is an embedded Lie subgroup, so a compact connected abelian subgroup of $G$ is a compact connected abelian Lie group, i.e. a torus; a torus of $G$ is by definition such a closed Lie subgroup (Cartan closed subgroup theorem, Tori and maximal tori).

[L3]

Every torus of a compact Lie group is contained in a maximal torus (Existence of maximal tori).

Proof

technique · direct

1.1

Since $G$ is Hausdorff and $A$ is compact, $A$ is closed by [L1]; by [L2] the subgroup $A$ is an embedded Lie subgroup, and it is a compact connected abelian Lie group, hence a torus in the sense of the definition.

L1L2

2.1

By [L3] the torus $A$ is contained in a maximal torus of $G$ , which is the required conclusion. The Axiom of Choice entered through the closed-subgroup theorem in [L2] and supplies the retained hypothesis of [L3]; the proof of [L3] itself uses no choice.

A1L2L3step 1.1∎

Depends on

Used by

Analytic and root-system Weyl groups agree Theorem

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)