# Cycle notation

{{#invoke:Hatnote|hatnote}} {{ safesubst:#invoke:Unsubst||$N=Merge to |date=__DATE__ |$B= Template:MboxTemplate:DMCTemplate:Merge partner }}

In mathematics, **cycle notation** is a useful convention for writing down a permutation in terms of its constituent cycles.^{[1]} This had sometimes been called **circular notation** and a permutation consisting of a single cycle was called a **circular** permutation.^{[2]} Modern terminology uses the term **cyclic** to mean a permutation with one cycle or one set of non-fixed points, and restricts a circular permutation to mean a permutation of objects arranged in a circle up to cyclic permutations, so that there is no fixed starting object on the circle.^{[3]}

## Definition

be distinct elements of . The expression

denotes the cycle σ whose action is

For each index *i*,

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

A 1-element cycle such as (3) is the identity permutation.^{[4]} The identity permutation can also be written as the empty cycle, "()".^{[5]}

## Permutation as product of cycles

Let be a permutation of , and let

be the orbits of . For an orbit , let denote the cardinality of . Also, choose an element , and define

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

In such an expression, it is typical, but not necessary, to suppress the 1-cycles.^{[6]} Thus, the permutation σ = (2 4 5)(1 6)(3) would be written as (2 4 5)(1 6) provided that it is understood that σ acts on *S*= {1,...,6}.

Since disjoint cycles commute with each other, the meaning of this expression is independent of the convention used for the order in products of permutations, namely whether the factors in such a product operate rightmost-first (as is usual more generally for function composition), or leftmost-first as some authors prefer. The meaning of individual cycles is also independent of this convention, namely always as described above.

## Example

Here are the 24 elements of the symmetric group on expressed using the cycle notation, and grouped according to their conjugacy classes:

## See also

## Notes

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

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