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# Cycle notation

In combinatorial mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles.

## Definition

Let $S$ be a finite set, and
$a_1,ldots,a_k,quad kgeq 2$

be distinct elements of $S$. The expression

$\left(a_1 ldots a_k\right)$

denotes the cycle σ whose action is

$a_1mapsto a_2mapsto a_3ldots a_k mapsto a_1.$
For each index i,
$sigma \left(a_i\right) = a_\left\{i+1\right\}.$

There are $k$ different expressions for the same cycle; the following all represent the same cycle:

$\left(a_1 a_2 a_3 ldots a_k\right) = \left(a_2 a_3 ldots a_k a_1\right) = cdots = \left(a_k a_1 a_2 ldots a_\left\{k-1\right\}\right).,$

A 1-element cycle is the same thing as the identity permutation and is omitted. It is customary to express the identity permutation simply as $\left(\right),$.

## Permutation as product of cycles

Let $pi$ be a permutation of $S$, and let

$S_1,ldots, S_ksubset S,quad kinmathbb\left\{N\right\}$

be the orbits of $pi$ with more than 1 element. For each $j=1,ldots,k$ let $n_j$ denote the cardinality of $S_j$. Also, choose an $a_\left\{1,j\right\}in S_j$, and define

$a_\left\{i+1,j\right\} = pi\left(a_\left\{i,j\right\}\right),quad iinmathbb\left\{N\right\}.,$

We can now express $pi$ as a product of disjoint cycles, namely

$pi = \left(a_\left\{1,1\right\} ldots a_\left\{n_1,1\right\}\right) \left(a_\left\{2,1\right\} ldots a_\left\{n_2,2\right\}\right) ldots \left(a_\left\{k,1\right\} ldots a_\left\{n_k,k\right\}\right).,$

## Example

There are the 24 elements of the symmetric group on $\left\{1,2,3,4\right\}$ expressed using the cycle notation, and grouped according to their conjugacy classes:

$\left(\right),$
$\left(12\right), ;\left(13\right),; \left(14\right),; \left(23\right),; \left(24\right),; \left(34\right)$ (transpositions)
$\left(123\right),; \left(132\right),; \left(124\right),; \left(142\right),; \left(134\right),; \left(143\right),; \left(234\right),; \left(243\right)$
$\left(12\right)\left(34\right),;\left(13\right)\left(24\right),; \left(14\right)\left(23\right)$
$\left(1234\right),; \left(1243\right),; \left(1324\right),; \left(1342\right),; \left(1423\right),; \left(1432\right)$