I'm a bit confused on the use of cycle notation
on this website https://groupprops.subwiki.org/wiki/Element_structure_of_symmetric_group:S4
it has the identity relation () in one line notation as (1 2 3 4)
But doesn't (1 2 3 4) imply that 1 maps to 2, that 2 maps to 3 and that 3 maps to 4 and 4 maps to 1?
How does that signify the identity relation?
The one-line notation (without parentheses!) is a compact notation for the more traditional two-line notation: \begin{alignat}{2} \text{one-line notation}&&&\text{cycle notation}\\ \begin{matrix}1&2&3&4\end{matrix}&\longleftrightarrow\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}& {}\longleftrightarrow{}&()\\ \begin{matrix}1&2&4&3\end{matrix}&\longleftrightarrow \begin{pmatrix}1&2&3&4\\1&2&4&3\end{pmatrix}& {}\longleftrightarrow{}& (34)\\ \begin{matrix}4&3&2&1\end{matrix}&\longleftrightarrow \begin{pmatrix}1&2&3&4\\4&3&2&1\end{pmatrix}& {}\longleftrightarrow{}&(14)(23) \end{alignat}