\begin{definition}[Definition:Permutation on n Letters/Cycle Notation/Canonical Representation]
The permutation:
:$\begin{pmatrix}
  1 & 2 & 3 & 4 & 5 \\
  2 & 1 & 4 & 3 & 5 
\end{pmatrix}$
can be expressed in '''cycle notation''' as:
:$\begin{pmatrix} 1 & 2 \end{pmatrix} \begin{pmatrix} 3 & 4 \end{pmatrix}$
or as:
:$\begin{pmatrix} 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \end{pmatrix} \begin{pmatrix} 1 & 2 \end{pmatrix}$
or as:
:$\begin{pmatrix} 4 & 3 \end{pmatrix} \begin{pmatrix} 2 & 1 \end{pmatrix}$
etc.
However, only the first is conventional. This is known as the '''canonical representation'''.
\end{definition}