Permutation
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 of elements, this is referred to as "decomposition into disjoint cycles".

To write down the permutation σ {\displaystyle \sigma } in cycle notation, one proceeds as follows:

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 σ (x) = x {\displaystyle \sigma (x)=x}.

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 σ = 265431 {\displaystyle \sigma =265431} can be written in cycle notation as:

σ = (126) (35) (4) = (126) (35). {\displaystyle \sigma =(126)(35)(4)=(126)(35).}

This may be seen as the composition σ = κ 1 κ 2 {\displaystyle \sigma =\kappa _{1}\kappa _{2}} of cyclic permutations:

κ 1 = (126) = (126) (3) (4) (5), κ 2 = (35) = (35) (1) (2) (6). {\displaystyle \kappa _{1}=(126)=(126)(3)(4)(5),\quad \kappa _{2}=(35)=(35)(1)(2)(6).}

While permutations in general do not commute, disjoint cycles do; for example:

σ = (126) (35) = (35) (126). {\displaystyle \sigma =(126)(35)=(35)(126).}

Also, each cycle can be rewritten from a different starting point; for example,

σ = (126) (35) = (261) (53). {\displaystyle \sigma =(126)(35)=(261)(53).}

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,

σ − 1 = (A 2 (126) (35)) − 1 = (621) (53). {\displaystyle \sigma ^{-1}=\left({\vphantom {A^{2}}}(126)(35)\right)^{-1}=(621)(53).}
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