cycleCycle22002-02-19 22:37:232007-04-30 09:28:44Definition%\usepackage{graphicx}
%\usepackage{xypic}
\usepackage{bbm}
\newcommand{\Z}{\mathbbmss{Z}}
\newcommand{\C}{\mathbbmss{C}}
\newcommand{\R}{\mathbbmss{R}}
\newcommand{\Q}{\mathbbmss{Q}}
\newcommand{\mathbb}[1]{\mathbbmss{#1}}Let $S$ be a set. A \emph{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.