cycle

Let $S$ be a set. A cycle is a permutation (bijective function of a set onto itself) such that there exist distinct elements $a_{1},a_{2},\ldots,a_{k}$ of $S$ such that

f(a_{i})=a_{i+1}\qquad\mbox{and}\qquad f(a_{k})=a_{1}

that is

$\displaystyle f(a_{1})$	$\displaystyle=$	$\displaystyle a_{2}$
$\displaystyle f(a_{2})$	$\displaystyle=$	$\displaystyle a_{3}$
	$\displaystyle\vdots$
$\displaystyle f(a_{k})$	$\displaystyle=$	$\displaystyle a_{1}$

and $f(x)=x$ for any other element of $S$ .

This can also be pictured as

a_{1}\mapsto a_{2}\mapsto a_{3}\mapsto\cdots\mapsto a_{k}\mapsto a_{1}

and

x\mapsto x

for any other element $x\in S$ , where $\mapsto$ represents the action of $f$ .

One of the basic results on symmetric groups says that any finite permutation can be expressed as product of disjoint cycles.

Title	cycle
Canonical name	Cycle1
Date of creation	2013-03-22 12:24:23
Last modified on	2013-03-22 12:24:23
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	10
Author	yark (2760)
Entry type	Definition
Classification	msc 03-00
Classification	msc 05A05
Classification	msc 20F55
Related topic	Permutation
Related topic	SymmetricGroup
Related topic	Transposition
Related topic	Group
Related topic	Subgroup
Related topic	DihedralGroup
Related topic	CycleNotation
Related topic	PermutationNotation