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