Group theory and generalizations Theorem

Decomposition of a permutation into disjoint cycles

Builds on Symmetric group Cycle notation Disjoint cycles commute

Statement

Let $X$ be a finite set and let $\sigma \in \operatorname{Sym}(X)$ . Then $\sigma$ is a product of cycles of length at least $2$ whose sets of moved points are pairwise disjoint. This product is unique up to the order of the factors.

Remarks

The cycles come from the orbits of $\langle \sigma \rangle$ on $X$ : each orbit with more than one point gives one cycle. The order of the factors does not matter, because disjoint cycles commute. For example, the permutation $1 \mapsto 2$ , $2 \mapsto 1$ , $3 \mapsto 4$ , $4 \mapsto 5$ , $5 \mapsto 3$ in $S_5$ is $(1\ 2)(3\ 4\ 5)$ .

Reading path

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Cycle notation concept
Group concept
Symmetric group definition
Disjoint cycles commute statement
Decomposition of a permutation into disjoint cycles statement

Tag	`0YWDAZPRPQ`
URI	`m2://0YWDAZPRPQ`
Kind	statement, theorem
Writing	W0 stub
MSC	20B30
Status	This copy of the site has no build data. Run `m2 status 0YWDAZPRPQ`.
Uses	Symmetric group, Cycle notation, Disjoint cycles commute

Decomposition of a permutation into disjoint cycles

Statement

Remarks

Reading path

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