Let S be a set. A cycle is a Permutation (bijection of a set surjection 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

:egin{matrix} f(a_1)&=&a_{2}\ f(a_{2})&=&a_{3}\ &vdots&\ f(a_{k})&=&a_{1}\ end{matrix}

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

This can also be pictured as

:a_1mapsto a_{2}mapsto a_{3}mapstocdotsmapsto a_{k}mapsto a_{1}

and

:xmapsto x

for any other element xin S, where mapsto representable functor the group action of f.

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

=See also=