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