Cycle notation

For the cyclic decomposition of graphs, see Cycle decomposition (graph 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] This is also called circular notation and the permutation called a cyclic or circular permutation.^[2]

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

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

For each index i,

σ(a_i) = a_{i + 1},

where a_{k + 1} is taken to mean a₁.

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 such as (3) is the identity permutation.^[3] The identity permutation can also be written as an empty cycle, "()".^[4]

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. Consider an element S_j, $j=1,\ldots,k$, let n_j denote the cardinality of S_j, | S_j | =n_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

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

Note that the usual convention in cycle notation is to multiply from left to right (in contrast with composition of functions, which is normally done from right to left). For example, the product $(1\ 2)(2\ 3)$ is equal to $(1\ 3\ 2)$ not $(1\ 2\ 3)$.

Example

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:

$( )\,$

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

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

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

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

See also

• Cyclic permutation
• Cycles and fixed points

Notes

1. ^ Fraleigh 2002:89; Hungerford 1997:230
2. ^ Dehn 1930:19
3. ^ Hungerford 1997:231
4. ^ Johnson 2003:691

References

• Dehn, Edgar (1960) [1930], Algebraic Equations, Dover .
• Fraleigh, John (2003), A first course in abstract algebra (7th ed.), Addison Wesley, p. 88–90, ISBN 978-0201763904 .
• Hungerford, Thomas W. (1997), Abstract Algebra: An Introduction, Brooks/Cole, ISBN 978-0030105593 .
• Johnson, James L. (2003), Probability and Statistics for Computer Science, Wiley Interscience, ISBN 978-0471326724 .

This article incorporates material from cycle notation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.