Cycle (mathematics)

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{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_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 permutation can be expressed as product of disjoint cycles. Furthermore, the representation of a permutation as a product of disjoint cycles is unique up to the order of the cycles. (It can be easily seen that disjoint cycles commute.) It thus follows that the number of the cycles in such a representation and its parity are functions of the permutation.

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