Cycle notation: Information from Answers.com

This article is about a convention in group theory. For cycling terminology, see glossary of bicycling.

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

1 Definition
2 Permutation as product of cycles
3 Example
4 See also

Definition

Let $S$ be a finite set, and

$a_1,\ldots,a_k,\quad k\geq 2$

be distinct elements of $S$ . The expression

$(a_1\ \ldots\ a_k)$

denotes the cycle σ whose action is

$a_1\mapsto a_2\mapsto a_3\ldots a_k \mapsto a_1.$

For each index i,

σ(a i) = a i + 1,

where $a k + 1$ is taken to mean $a 1$ .

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

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

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 $()\,$ .

Permutation as product of cycles

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

$S_1,\ldots, S_k\subset S,\quad k\in\mathbb{N}$

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

$a_{i+1,j} = \pi(a_{i,j}),\quad i\in\mathbb{N}.\,$

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

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

Example

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

$( )\,$

$(1 2), \;(1 3),\; (1 4),\; (2 3),\; (2 4),\; (3 4)$ (transpositions)

$(1 2 3),\; (1 3 2),\; (1 2 4),\; (1 4 2),\; (1 3 4),\; (1 4 3),\; (2 3 4),\; (2 4 3)$

$(1 2)(3 4),\;(1 3)(2 4),\; (1 4)(2 3)$

$(1 2 3 4),\; (1 2 4 3),\; (1 3 2 4),\; (1 3 4 2),\; (1 4 2 3),\; (1 4 3 2)$

	Transposition (mathematics)
	List of permutation topics
	Cycle (mathematics)

	Example program of how to convert infix notation to postfix notation and prefix notation? Read answer...
	What is exponential notation? Read answer...
	What is the notation for a ray? Read answer...

	What is an orbital notation?
	What is a fractional notation?
	What are legal notations?

Cycle notation

Contents

Definition

Permutation as product of cycles

Example

See also