Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-09
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A group is abelian exactly when its conjugacy classes are singletons

Statement

A group G is abelian if and only if every conjugacy class in G is a singleton. No finiteness assumption is needed.

Facts & Assumptions

Given: A group G, with identity e.

[F1]

Cl⁡G(h)={ghg−1:g∈G} (The conjugacy class Cl⁡G(x) and centralizer CG(x) of an element).

Proof

technique · direct
1.1

If G is abelian, then for all g,h∈G one has ghg−1=hgg−1=h. Thus every element of Cl⁡G(h) is h; and ehe−1=h shows that h belongs to the class. Hence Cl⁡G(h)={h}.

F1givenalgebra
1.2

Conversely, suppose every class is a singleton. Since ehe−1=h, that singleton must be {h}. For arbitrary g,h, F1 then gives ghg−1=h. Multiplying on the right by g yields gh=hg. Thus every pair commutes and G is abelian.

F1algebra
2.1

In the trivial group the sole class is {e} and ee=ee, so both properties hold. In all groups, the membership h∈Cl⁡G(h) used above prevents empty classes. The argument quantifies over arbitrary g,h and makes no cardinality assumption.

F1step 1.1step 1.2∎

Sources

Judson, §14.2 opening identifies fixed points of conjugation with the center. Etingof et al., §4.3(1), p. 64, uses singleton classes for abelian groups. Both implications are proved above without finiteness.

Depends on

Used by

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Dependency tree · two levels

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Sources