Compose permutations clearly. View cycles, inverses, parity, and order. Export results quickly for lessons, homework, exams, and verification.
Choose the size n, then enter σ(i) and τ(i). Each row must contain a valid permutation of 1 through n.
| i | σ(i) | τ(i) | (σ ∘ τ)(i) | (τ ∘ σ)(i) |
|---|---|---|---|---|
| 1 | 2 | 3 | 1 | 1 |
| 2 | 3 | 1 | 2 | 5 |
| 3 | 1 | 2 | 3 | 2 |
| 4 | 5 | 4 | 5 | 4 |
| 5 | 4 | 5 | 4 | 3 |
This example shows how composition order changes the outcome. In many cases, σ ∘ τ and τ ∘ σ are different.
Composition rule: For permutations σ and τ on the same set, the composition is
(σ ∘ τ)(i) = σ(τ(i))
This means apply τ first, then apply σ to the result.
Inverse rule:
σ⁻¹(σ(i)) = i
Order of a permutation: Convert the permutation into disjoint cycles. Then compute the least common multiple of the cycle lengths.
order(σ) = lcm(length of each disjoint cycle)
Parity: A permutation is even or odd depending on whether it can be written as an even or odd number of transpositions.
Permutation composition means applying one permutation after another. If you compute σ ∘ τ, you first apply τ to an element, then apply σ to that output.
Order matters because permutation composition is usually not commutative. In general, σ ∘ τ and τ ∘ σ send elements to different outputs, so both are worth checking.
Cycle notation groups elements that map around in loops. It is compact, easier to interpret, and useful for finding order, inverse behavior, and structural properties.
The order is the smallest positive integer k such that σ^k becomes the identity. It equals the least common multiple of the lengths of disjoint cycles.
A fixed point is an element i for which σ(i) = i. Fixed points stay unchanged under the permutation and help reveal identity-like behavior.
Parity tells whether a permutation can be built from an even or odd number of swaps. This property is important in algebra, determinants, and sign calculations.
No. A valid permutation must use every integer from 1 through n exactly once. Repeated or missing values make the row invalid.
The graph compares how each element maps under σ, τ, σ ∘ τ, and τ ∘ σ. It gives a quick visual check of how the outputs differ.
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