Permutations
Composition is right to left: (σ ∘ τ)(x) = σ(τ(x)). In σ ∘ τ, τ acts first and σ second. A product of cycles such as (1 2 3)(3 4) is read the same way.
Leave empty to use the largest element that appears. Set it to act on a larger set: the extra elements are fixed points.
Permutation σ
Permutation τ (optional)
Result
- Disjoint cycles
- One-line
- Two-line
Both compositions
Calculation table for σ ∘ τ
Each row follows one element: first through τ, then through σ.
| x | τ(x) | σ(τ(x)) |
|---|
Analysis
Order
Sign, parity and inversions
Fixed points and support
Product of transpositions
Advanced properties: centralizer and conjugacy class
Visualizations
Mapping diagram
Cycle diagram
About this calculator
A permutation and its notations. A permutation of {1, …, n} is a rule that sends each element to exactly one element, with no two sent to the same place. The three notations are three ways of writing that one rule. One-line notation lists the images in order: [2, 3, 1, 4] means 1 → 2, 2 → 3, 3 → 1, 4 → 4. Two-line notation writes each element above its image. Cycle notation follows the elements round: (1 2 3) means 1 → 2 → 3 → 1, and everything not written stays where it is.
Right to left. Permutations are functions, and σ ∘ τ means “first τ, then σ”, exactly as f(g(x)) applies g first. With σ = (1 2 3) and τ = (1 2): τ sends 1 to 2 and σ sends 2 to 3, so σ ∘ τ sends 1 to 3, and altogether σ ∘ τ = (1 3) while τ ∘ σ = (2 3). The order matters. Some books compose left to right; their products are the ones this page calls τ ∘ σ.
Products of cycles. A cycle expression such as (1 2 3)(3 4) is a product of two permutations and is evaluated by the same rule: (3 4) first, then (1 2 3), which gives (1 2 3 4). The cycles you type may share elements. The answer is always shown as disjoint cycles, each starting with its smallest element, and that form is unique. So a permutation has many cycle expressions but only one canonical one.
Degree and fixed points. The degree n says which set is being permuted. One-line and two-line notation fix it by their length. Cycle notation does not: (1 2 3) could act on three elements or on ten. Left alone, the calculator uses the largest element it sees; set n to say otherwise, or write a fixed point as a 1-cycle, as in (1 2 3)(7). Fixed points are left out of cycle notation unless you ask to see them, but the degree is never forgotten: it changes the cycle type, the sign formula and the sizes of the centralizer and the conjugacy class. The identity is written id.
Order and sign. The order of σ is the smallest positive k with σ^k = id. A cycle of length m returns to the start after m steps, so the order is the least common multiple of the cycle lengths. The sign is (−1)^(n − c) with c the number of cycles, fixed points included; it is +1 when σ is a product of an even number of transpositions and −1 when the number is odd, and it also equals (−1)^inv(σ).
Limits. Everything is computed in your browser, exactly, for degrees up to 5,000; nothing is ever done by listing all n! permutations. The diagrams are drawn for small degrees only, and long results are shortened on screen (Copy always gives the whole value).