Editing Permutation (section)

===Cycle notation===
Cycle notation describes the effect of repeatedly applying the permutation on the elements of the set ''S'', with an orbit being called a ''cycle''. The permutation is written as a list of cycles; since distinct cycles involve [[disjoint sets|disjoint]] sets of elements, this is referred to as "decomposition into disjoint cycles".

To write down the permutation <math>\sigma</math> in cycle notation, one proceeds as follows:

# Write an opening bracket followed by an arbitrary element ''x'' of <math>S</math>:  <math>(\,x</math>
# Trace the orbit of ''x'', writing down the values under successive applications of <math>\sigma</math>: <math>(\,x,\sigma(x),\sigma(\sigma(x)),\ldots</math>
# Repeat until the value returns to ''x,'' and close the parenthesis without repeating ''x'': <math>(\,x\,\sigma(x)\,\sigma(\sigma(x))\,\ldots\,)</math>
# Continue with an element ''y'' of ''S'' which was not yet written, and repeat the above process: <math>(\,x\,\sigma(x)\,\sigma(\sigma(x))\,\ldots\,)(\,y\,\ldots\,)</math>
# Repeat until all elements of ''S'' are written in cycles.

Also, it is common to omit 1-cycles, since these can be inferred: for any element ''x'' in ''S'' not appearing in any cycle, one implicitly assumes <math>\sigma(x) = x</math>.<ref>{{harvnb|Hall|1959|p=54}}</ref>

Following the convention of omitting 1-cycles, one may interpret an individual cycle as a permutation which fixes all the elements not in the cycle (a [[cyclic permutation]] having only one cycle of length greater than 1). Then the list of disjoint cycles can be seen as the composition of these cyclic permutations. For example, the one-line permutation <math>\sigma = 2 6 5 4 3 1  </math> can be written in cycle notation as:<blockquote><math>\sigma = (126)(35)(4) = (126)(35).  </math></blockquote>This may be seen as the composition <math>\sigma = \kappa_1 \kappa_2  </math> of cyclic permutations:<blockquote><math>\kappa_1 = (126) = (126)(3)(4)(5),\quad \kappa_2 = (35)= (35)(1)(2)(6).  </math> </blockquote>While permutations in general do not commute, disjoint cycles do; for example:<blockquote><math>\sigma = (126)(35) = (35)(126).  </math></blockquote>Also, each cycle can be rewritten from a different starting point; for example,<blockquote><math>\sigma = (126)(35) = (261)(53).  </math></blockquote>Thus one may write the disjoint cycles of a given permutation in many different ways.

A convenient feature of cycle notation is that inverting the permutation is given by reversing the order of the elements in each cycle. For example, <blockquote><math>\sigma^{-1} = \left(\vphantom{A^2}(126)(35)\right)^{-1} = (621)(53).  </math></blockquote>