ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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.

The seven conjugacy classes of $S_{5}$ and their centralizer and class sizes

Example

The conjugacy data for $S_{5}$ are

cycle type	centralizer size	class size	parity	splits in $A_{5}$ ?
$1^{5}$	$120$	$1$	even	no
$2, 1^{3}$	$12$	$10$	odd	--
$2^{2}, 1$	$8$	$15$	even	no
$3, 1^{2}$	$6$	$20$	even	no
$3, 2$	$6$	$20$	odd	--
$4, 1$	$4$	$30$	odd	--
$5$	$5$	$24$	even	yes

Facts & Assumptions

Given: The symmetric group $S_{5}$ .

[F1]

A cycle type $(c_{k})$ has centralizer size $\prod k^{c_{k}} c_{k}!$ (If $σ \in S_{n}$ has $c_{k}$ cycles of length $k$ , then $∣ C_{S_{n}} (σ) ∣ = \prod_{k = 1}^{n} k^{c_{k}} c_{k}!$ ).

[F2]

$S_{n}$ -classes are indexed by the tuples with $\sum k c_{k} = n$ (The conjugacy classes of $S_{n}$ are indexed by the tuples $(c_{1}, \dots, c_{n})$ with $\sum k c_{k} = n$ ), and $n! = \sum n! / \prod_{k} k^{c_{k}} c_{k}!$ over those tuples, the summand indexed by $(c_{k})$ being the size of the corresponding class (The class equation of $S_{n}$ is $n! = \sum_{\sum k c_{k} = n} n! / \prod_{k} k^{c_{k}} c_{k}!$ ).

[F3]

A $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (σ) = (- 1)^{n - c (σ)}$ when fixed points are included as $1$ -cycles (A $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (σ) = (- 1)^{n - c (σ)}$ when fixed points are counted as cycles).

[F4]

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).

Verification

technique · counting

1.1

The seven partitions of $5$ give the seven rows. Applying [F1] gives the centralizer column, and [F2] gives the class-size column.

F1F2algebra

2.1

The sizes sum to $1 + 10 + 15 + 20 + 20 + 30 + 24 = 120$ , verifying the class equation.

F2step 1.1algebra

3.1

Formula [F3] gives the parity column. Among the even rows, [F4] applies only to the single cycle of length $5$ , giving the final column.

F3F4step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

K. Conrad, Conjugacy Classes (standard reference, not scraped)

The seven conjugacy classes of S5S5​ and their centralizer and class sizes