Group theory and generalizations Theorem

Conjugacy classes of the symmetric group correspond to cycle types

Builds on Cycle type Conjugate elements Symmetric group

Statement

Let $\sigma, \tau \in S_n$ . Then $\sigma$ and $\tau$ are conjugate in $S_n$ if and only if they have the same cycle type.

Remarks

The key formula is $\pi (a_1\ a_2\ \dots\ a_k) \pi^{-1} = (\pi(a_1)\ \pi(a_2)\ \dots\ \pi(a_k))$ for each $\pi \in S_n$ . So conjugation keeps the cycle type, and a suitable $\pi$ sends the cycles of $\sigma$ onto the cycles of $\tau$ . Thus the number of conjugacy classes of $S_n$ is the number of partitions of $n$ . For example, $S_4$ has $5$ conjugacy classes.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Cycle notation concept
Group concept
Conjugate elements definition
Symmetric group definition
Disjoint cycles commute statement
Decomposition of a permutation into disjoint cycles statement
Cycle type definition
Conjugacy classes of the symmetric group correspond to cycle types statement

Tag	`0J438GAC8K`
URI	`m2://0J438GAC8K`
Kind	statement, theorem
Writing	W0 stub
MSC	20B30
Status	This copy of the site has no build data. Run `m2 status 0J438GAC8K`.
Uses	Cycle type, Conjugate elements, Symmetric group

Conjugacy classes of the symmetric group correspond to cycle types

Statement

Remarks

Reading path

Further reading

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