Group theory and generalizations Proposition

Disjoint cycles commute

Builds on Cycle notation Symmetric group

Statement

Let $\sigma$ and $\tau$ be cycles in the symmetric group $\operatorname{Sym}(X)$ whose sets of moved points are disjoint. Then $\sigma \tau = \tau \sigma$ .

Remarks

A point moved by $\sigma$ is fixed by $\tau$ , and $\sigma$ maps it to another point that $\tau$ fixes. So both products send it to the same point. The same argument works for the points moved by $\tau$ , and the points fixed by both are fixed by both products. Cycles that are not disjoint need not commute, for example $(1\ 2)(1\ 3) \neq (1\ 3)(1\ 2)$ .

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Cycle notation concept
Group concept
Symmetric group definition
Disjoint cycles commute statement

Tag	`07E6D8R4D6`
URI	`m2://07E6D8R4D6`
Kind	statement, proposition
Writing	W0 stub
MSC	20B30
Status	This copy of the site has no build data. Run `m2 status 07E6D8R4D6`.
Uses	Cycle notation, Symmetric group

Disjoint cycles commute

Statement

Remarks

Reading path

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