Editing Cyclic permutation (section)

== Basic properties ==
{{Hatnote|In the remainder of the article, the enlarged definition is used.}}
One of the basic results on [[symmetric group]]s is that any permutation can be expressed as the product of [[Disjoint sets|disjoint]] cycles (more precisely: cycles with disjoint orbits); such cycles commute with each other, and the expression of the permutation is unique up to the order of the cycles.{{efn|Note that the cycle notation is not unique: each ''k''-cycle can itself be written in ''k'' different ways, depending on the choice of <math>s_0</math> in its orbit.}} The [[multiset]] of lengths of the cycles in this expression (the [[cycle type]]) is therefore uniquely determined by the permutation, and both the signature and the [[conjugacy class]] of  the permutation in the symmetric group are determined by it.<ref>{{harvnb|Rotman|2006|loc=p. 117, 121}}</ref>

The number of ''k''-cycles in the symmetric group ''S''<sub>''n''</sub> is given, for <math>1\leq k\leq n</math>, by the following equivalent formulas:
<math display="block">\binom nk(k-1)!=\frac{n(n-1)\cdots(n-k+1)}{k}=\frac{n!}{(n-k)!k}.</math>

A ''k''-cycle has [[signature of a permutation|signature]] (−1)<sup>''k''&nbsp;−&nbsp;1</sup>.

The [[inverse function|inverse]] of a cycle <math>\sigma = (s_0~s_1~\dots~s_{k-1})</math> is given by reversing the order of the entries: <math>\sigma^{-1} = (s_{k - 1}~\dots~s_1~s_{0})</math>.  In particular, since <math>(a ~ b) = (b ~ a)</math>, every two-cycle is its own inverse.  Since disjoint cycles commute, the inverse of a product of disjoint cycles is the result of reversing each of the cycles separately.