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$S_{4}$ has five conjugacy classes of sizes $1$ , $6$ , $8$ , $6$ , and $3$

Example

The group $S_{4}$ has five conjugacy classes, represented by $1$ , $(12)$ , $(123)$ , $(1234)$ , and $(12) (34)$ , with sizes $1$ , $6$ , $8$ , $6$ , and $3$ respectively.

Facts & Assumptions

Given: The symmetric group $S_{4}$ .

[F1]

The conjugacy classes of $S_{n}$ are indexed by cycle types (The conjugacy classes of $S_{n}$ are indexed by the tuples $(c_{1}, \dots, c_{n})$ with $\sum k c_{k} = n$ ).

[F2]

The class equation counts permutations by cycle type: $n! = \sum_{c} \frac{n!}{\prod_{k} k^{c_{k}} c_{k}!}$ over all tuples $c$ with $\sum_{k} k c_{k} = n$ (The class equation of $S_{n}$ is $n! = \sum_{\sum k c_{k} = n} n! / \prod_{k} k^{c_{k}} c_{k}!$ ).

[A1]

The cycle types of elements of $S_{4}$ are $(4)$ , $(3, 1)$ , $(2, 2)$ , $(2, 1, 1)$ , and $(1, 1, 1, 1)$ .

Verification

technique · direct

1.1

By [F1] and [A1], the classes of $S_{4}$ are exactly those five cycle types.

F1A1given

1.2

The class size of a cycle type $c$ is $\frac{4!}{\prod_{k} k^{c_{k}} c_{k}!}$ by [F2]. For the types of [A1] these sizes are $\frac{24}{4} = 6$ , $\frac{24}{3} = 8$ , $\frac{24}{2^{2} \cdot 2} = 3$ , $\frac{24}{2} = 6$ , and $1$ .

F2A1algebra

2.1

Matching the representatives $1$ , $(12)$ , $(123)$ , $(1234)$ , $(12) (34)$ with their cycle types in [A1] and the corresponding values from step 1.2 gives class sizes $1$ , $6$ , $8$ , $6$ , $3$ . The sum $1 + 6 + 8 + 6 + 3 = 24 = ∣ S_{4} ∣$ , so the count is complete.

A1step 1.1step 1.2algebra∎

Depends on

Used by

The character table of S₄ and the normal subgroups it reveals Example

Dependency tree · two levels

9 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

Peter Webb, A Course in Finite Group Representation Theory, Example 3.3.5 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Example 3.15 (standard reference, not scraped)

S4S4​ has five conjugacy classes of sizes 11, 66, 88, 66, and 33