Cyclic permutation

In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set "X" which maps the elements of some subset "S" of "X" to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of "X". If "S" has "k" elements, the cycle is called a "k"-cycle. Cycles are often denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted.
For example, given "X" = {1, 2, 3, 4}, the permutation (1, 3, 2, 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 (so "S" = "X") is a 4-cycle, and the permutation (1, 3, 2) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 (so "S" = {1, 2, 3} and 4 is a fixed element) is a 3-cycle. On the other hand, the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs {1, 3} and {2, 4}.
The set "S" is called the orbit of the cycle. Every permutation on finitely many elements can be decomposed into cycles on disjoint orbits.
The cyclic parts of a permutation are cycles, thus the second example is composed of a 3-cycle and a 1-cycle (or "fixed point") and the third is composed of two 2-cycles, and denoted (1, 3) (2, 4).
Definition.
A permutation is called a cyclic permutation if and only if it has a single nontrivial cycle (a cycle of length > 1).
For example, the permutation, written in two-line notation (in two ways) and also cycle notations,
is a six-cycle; its cycle diagram is shown at right.
Some authors restrict the definition to only those permutations which consist of one nontrivial cycle (that is, no fixed points allowed).
For example, the permutation
is a cyclic permutation under this more restrictive definition, while the preceding example is not.
More formally, a permutation formula_3 of a set "X", viewed as a bijective function formula_4, is called a cycle if the action on "X" of the subgroup generated by formula_3 has at most one orbit with more than a single element. This notion is most commonly used when "X" is a finite set; then of course the largest orbit, "S", is also finite. Let formula_6 be any element of "S", and put formula_7 for any formula_8. If "S" is finite, there is a minimal number formula_9 for which formula_10. Then formula_11, and formula_3 is the permutation defined by
and formula_14 for any element of formula_15. The elements not fixed by formula_3 can be pictured as
A cycle can be written using the compact cycle notation formula_18 (there are no commas between elements in this notation, to avoid confusion with a "k"-tuple). The "length" of a cycle is the number of elements of its largest orbit. A cycle of length "k" is also called a "k"-cycle.
The orbit of a 1-cycle is called a "fixed point" of the permutation, but as a permutation every 1-cycle is the identity permutation. When cycle notation is used, the 1-cycles are often suppressed when no confusion will result.
Basic properties.
One of the basic results on symmetric groups is that any permutation can be expressed as the product of 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. 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.
The number of "k"-cycles in the symmetric group "S""n" is given, for formula_19, by the following equivalent formulas
A "k"-cycle has signature (−1)"k" − 1.
The inverse of a cycle formula_18 is given by reversing the order of the entries: formula_22. In particular, since formula_23, 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.
Transpositions.
A cycle with only two elements is called a transposition. For example, the permutation formula_24 that swaps 2 and 4.
Properties.
Any permutation can be expressed as the composition (product) of transpositions—formally, they are generators for the group. In fact, when the set being permuted is for some integer , then any permutation can be expressed as a product of formula_25 and so on. This follows because an arbitrary transposition can be expressed as the product of adjacent transpositions. Concretely, one can express the transposition formula_26 where formula_27 by moving to one step at a time, then moving back to where was, which interchanges these two and makes no other changes:
The decomposition of a permutation into a product of transpositions is obtained for example by writing the permutation as a product of disjoint cycles, and then splitting iteratively each of the cycles of length 3 and longer into a product of a transposition and a cycle of length one less:
This means the initial request is to move formula_30 to formula_31 formula_32 to formula_33 formula_34 to formula_35 and finally formula_36 to formula_37 Instead one may roll the elements keeping formula_30 where it is by executing the right factor first (as usual in operator notation, and following the convention in the article on Permutations). This has moved formula_36 to the position of formula_31 so after the first permutation, the elements formula_30 and formula_36 are not yet at their final positions. The transposition formula_43 executed thereafter, then addresses formula_36 by the index of formula_32 to swap what initially were formula_30 and formula_47
In fact, the symmetric group is a Coxeter group, meaning that it is generated by elements of order 2 (the adjacent transpositions), and all relations are of a certain form.
One of the main results on symmetric groups states that either all of the decompositions of a given permutation into transpositions have an even number of transpositions, or they all have an odd number of transpositions. This permits the parity of a permutation to be a well-defined concept.