Cyclic Permutation/Examples

Examples of Cyclic Permutations

Symmetric Group on $3$ Letters

Each of the non-identity elements of the Symmetric Group on 3 Letters is a cyclic permutation.

Expressed in cycle notation, they are as follows:

\(\ds e\)

\(:=\)

\(\ds \text { the identity mapping}\)

\(\ds p\)

\(:=\)

\(\ds \tuple {1 2 3}\)

\(\ds q\)

\(:=\)

\(\ds \tuple {1 3 2}\)

\(\ds r\)

\(:=\)

\(\ds \tuple {2 3}\)

\(\ds s\)

\(:=\)

\(\ds \tuple {1 3}\)

\(\ds t\)

\(:=\)

\(\ds \tuple {1 2}\)

Non-Cyclic Element of Symmetric Group on $4$ Letters

Not all permutations are cycles.

Here is an example (written in two-row notation) of a permutation of the Symmetric Group on 4 Letters which is not a cycle:

$\begin{pmatrix}

1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{pmatrix}$