Cycle notation: Difference between revisions
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Revision as of 01:16, 22 April 2011
This article relies largely or entirely on a single source. (April 2011) |
In combinatorial mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles.
Definition
be distinct elements of . The expression
denotes the cycle σ whose action is
For each index i,
where is taken to mean .
There are different expressions for the same cycle; the following all represent the same cycle:
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 .[citation needed]
Permutation as product of cycles
Let be a permutation of , and let
be the orbits of with more than 1 element. Consider an element , , let denote the cardinality of , =. Also, choose an , and define
We can now express as a product of disjoint cycles, namely
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 is equal to not .
Example
There are the 24 elements of the symmetric group on expressed using the cycle notation, and grouped according to their conjugacy classes:
See also
References
- Fraleigh, John (2002), A first course in abstract algebra (7th ed.), Addison Wesley, ISBN 978-0201763904