LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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 $α$ and $β$ are cycles in $Sym (X)$ and $supp (α) \cap supp (β) = \emptyset$ , then $α β = β α$ .

Facts & Assumptions

Given: A finite set $X$ and two cycles $α, β \in Sym (X)$ 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

1.1givenL1

Every $x \in X$ lies in $supp (α)$ , in $supp (β)$ , or in neither support, and the first two alternatives cannot both hold.

2.1step 1.1L1∎

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

Depends on

Dependency tree · 2 levels

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, §5.1, Proposition 5.2