PlanetMath: cycle

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cycle

(Definition)

Let 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 such that

$\begin{displaymath}f(a_i) = a_{i+1}\qquad \mbox{and}\qquad f(a_k)=a_1\end{displaymath}$

that is

$\begin{eqnarray*} f(a_1)&=&a_{2}\ f(a_{2})&=&a_{3}\ &\vdots&\ f(a_{k})&=&a_{1}\ \end{eqnarray*}$

and

for any other element of

This can also be pictured as

$\begin{displaymath}a_1\mapsto a_{2}\mapsto a_{3}\mapsto\cdots\mapsto a_{k}\mapsto a_{1}\end{displaymath}$

and

$\begin{displaymath}x\mapsto x\end{displaymath}$

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

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

"cycle" is owned by yark. [ full author list (2) | owner history (2) ]

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Attachments:: every permutation has a cycle decomposition (Proof) by rspuzio

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Cross-references: disjoint, product, finite, symmetric groups, action, represents, onto, bijective function, permutation
There are 23 references to this entry.

This is version 7 of cycle, born on 2002-02-19, modified 2007-04-30.
Object id is 2262, canonical name is Cycle2.
Accessed 4557 times total.

Classification:

AMS MSC:	20F55 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Reflection and Coxeter groups)
	05A05 (Combinatorics :: Enumerative combinatorics :: Combinatorial choice problems )
	03-00 (Mathematical logic and foundations :: General reference works )

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