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 groupssays 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