LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How statement and proof provenance work

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.

Cycles with disjoint supports commute

Statement

Cycles with disjoint supports commute. More precisely, if $X$ is finite and $\alpha$ and $\beta$ are cycles in $\operatorname{Sym}(X)$ and $\operatorname{supp}(\alpha)\cap\operatorname{supp}(\beta)=\varnothing$ , then $\alpha\beta=\beta\alpha$ .

Facts & Assumptions

Given: A finite set $X$ and two cycles $\alpha,\beta\in\operatorname{Sym}(X)$ with disjoint supports.

[L1]

A cycle fixes every point outside its support, and two cycles are disjoint exactly when their supports are disjoint (Support, fixed points, disjoint cycles, cycle length, disjoint-cycle decompositions, and cycle type).

Proof

technique · direct

1.1

Every $x\in X$ lies in $\operatorname{supp}(\alpha)$ , in $\operatorname{supp}(\beta)$ , or in neither support, and the first two alternatives cannot both hold.

givenL1

2.1

If $x\in\operatorname{supp}(\alpha)$ , then $\beta$ fixes both $x$ and $\alpha(x)$ , so $\alpha\beta(x)=\beta\alpha(x)=\alpha(x)$ ; the symmetric argument applies on $\operatorname{supp}(\beta)$ , while outside both supports both cycles fix $x$ . Thus the two composites agree at every point.

step 1.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, §5.1, Proposition 5.2 (standard reference, not scraped)