Definition:Permutation on n Letters/Cycle Notation/Canonical Representation

Definition

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.

Sources

• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory