Canonical cycle notation of a permutation

Dependencies:

1. Product of disjoint cycles is commutative
2. Permutation is disjoint cycle product

Canonical cycle notation is a unique way of writing a permutation as a product of disjoint cycles.

This works when the set on which the permutation is defined is orderable.

The canonical representation of a cycle is when the smallest element of the cycle is written as the first element of the cycle.

A permutation can be written as a product of a unique set of disjoint cycles. Sort the cycles based on the smallest element of the cycles. Then write each cycle in its canonical form.

The permutation $(1, 2, 5)(3, 6, 4)$ is in canonical cycle notation.

The permutations $(3, 6, 4)(1, 2, 5)$ and $(1, 2, 5)(6, 4, 3)$ are not in canonical cycle notation.

Dependency for:

1. Parity of a permutation

Info:

• Depth: 4
• Number of transitive dependencies: 7

Transitive dependencies:

1. /sets-and-relations/relation-composition-is-associative
2. /sets-and-relations/composition-of-bijections-is-a-bijection
3. Group
4. Subgroup
5. Permutation group
6. Permutation is disjoint cycle product
7. Product of disjoint cycles is commutative