CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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.

All $3$ -cycles are conjugate in $A_{n}$ for $n \geq 5$

Statement

For $n \geq 5$ , all $3$ -cycles in $A_{n}$ lie in one $A_{n}$ -conjugacy class.

Facts & Assumptions

Given: $n \geq 5$ .

[F1]

Two permutations in $S_{n}$ are conjugate exactly when their cycle types agree (Two elements of $S_{n}$ are conjugate if and only if they have the same cycle type).

[F2]

For $n \geq 2$ and $σ \in A_{n}$ , the $S_{n}$ -class of $σ$ splits into two $A_{n}$ -classes of equal size exactly when all cycle lengths in its decomposition, including $1$ -cycles for fixed points, are odd and no two are equal (For $n \geq 2$ , an $S_{n}$ -class of an even permutation splits in $A_{n}$ exactly when all cycle lengths, including $1$ -cycles, are odd and distinct).

Proof

technique · direct

1.1

Every $3$ -cycle has cycle type consisting of one $3$ -cycle and $n - 3$ fixed points, so [F1] puts all of them in one $S_{n}$ -class.

2.1

Since $n - 3 \geq 2$ , the length $1$ is repeated. Thus [F2] says that this class does not split in $A_{n}$ .

F2step 1.1

3.1

Therefore all $3$ -cycles form one $A_{n}$ -conjugacy class.

step 1.1step 2.1∎

Depends on

Used by

A normal subgroup of Aₙ containing one 3-cycle equals Aₙ for n≥5 Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 30 results over 9 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. Judson, Abstract Algebra: Theory and Applications, Simplicity of $A_n$ (standard reference, not scraped)

All 33-cycles are conjugate in AnAn​ for n≥5n≥5