PlanetMath: cycle notation

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cycle notation

(Definition)

The cycle notation is a useful convention for writing down a permutations in terms of its constituent cycles. Let be a finite set, and

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

distinct elements of

. The expression $(a_1,\ldots,a_k)$ denotes the cycle whose action is

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

Note there are

different expressions for the same cycle; the following all represent the same cycle:

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

Also note that a 1-element cycle is the same thing as the identity permutation, and thus there is not much point in writing down such things. Rather, it is customary to express the identity permutation simply as

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

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

be the orbits of $\pi$ with more than 1 element. For each $j=1,\ldots,k$ let

denote the cardinality of

. Also, choose an $a_{1,j}\in S_j$ , and define

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

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

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

By way of illustration, here 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:

	$\displaystyle (),$
	$\displaystyle (12), \;(13),\; (14),\; (23),\; (24),\; (34)$
	$\displaystyle (123),\; (213),\; (124),\; (214),\; (134),\; (143),\; (234),\; (243)$
	$\displaystyle (12)(34),\;(13)(24),\; (14)(23)$
	$\displaystyle (1234),\; (1243),\; (1324),\; (1342),\; (1423),\; (1432)$

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"cycle notation" is owned by rmilson. [ full author list (2) ]

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Cross-references: conjugacy classes, symmetric group, disjoint, product, cardinality, orbits, point, identity, represent, action, expression, finite set, cycles, terms, permutations
There are 9 references to this entry.

This is version 3 of cycle notation, born on 2002-03-30, modified 2007-08-09.
Object id is 2808, canonical name is CycleNotation.
Accessed 5341 times total.

Classification:

AMS MSC:	20B05 (Group theory and generalizations :: Permutation groups :: General theory for finite groups)
	05A05 (Combinatorics :: Enumerative combinatorics :: Combinatorial choice problems )

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