Editing Symmetric group (section)

=== Multiplication ===
The group operation in a symmetric group is function composition, denoted by the symbol ∘ or by simple juxtaposition. The composition {{math|''f'' ∘ ''g''}} of permutations {{mvar|f}} and {{mvar|g}}, pronounced "{{mvar|f}} of {{mvar|g}}", maps any element {{mvar|x}} of {{mvar|X}} to {{math|''f''(''g''(''x''))}}. Concretely, let (see [[permutation]] for an explanation of notation):
<math display=block> f = (1~3)(2)(4~5)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end{pmatrix},</math>
<math display=block> g = (1~2~5)(3~4)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix}.</math>
Applying {{mvar|f}} after {{mvar|g}} maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So, composing {{mvar|f}} and {{mvar|g}} gives
<math display=block> fg = f\circ g = (1\ 2\ 4)(3\ 5)=\begin{pmatrix} 1 & 2 &3 & 4 & 5 \\ 2 & 4 & 5 & 1 & 3\end{pmatrix}.</math>

A [[Cyclic permutation|cycle]] of length {{math|''L'' {{=}} ''k'' · ''m''}}, taken to the {{mvar|k}}th power, will decompose into {{mvar|k}} cycles of length {{mvar|m}}: For example, ({{math|''k'' {{=}} 2}}, {{math|''m'' {{=}} 3}}),
<math display=block> (1~2~3~4~5~6)^2 = (1~3~5) (2~4~6).</math>