Results
Permutation Summary
| Name | One-Line | Cycle Notation | Order | Parity | Inversions | Fixed Points | Support | Cycle Type |
|---|
Generated Group Details
Mapping Graph
Calculator Inputs
Enter permutations in one-line form like 2 3 1 5 4 or cycle form like (1 2 3)(4 5). Leave permutation B blank to use the identity.
Example Data Table
| Example | Degree | Permutation A | Permutation B | Order(A) | Order(B) | Generated Group Size |
|---|---|---|---|---|---|---|
| Example 1 | 4 | (1 2)(3 4) | (1 2 3 4) | 2 | 4 | 8 |
| Example 2 | 4 | (1 2 3) | (1 2) | 3 | 2 | 6 |
| Example 3 | 5 | (1 2 3)(4 5) | (2 4)(3 5) | 6 | 2 | 120 |
Formula Used
1) Composition
For permutations σ and τ, the calculator uses (σ ∘ τ)(i) = σ(τ(i)). This means the right permutation acts first.
2) Inverse
If σ(i) = j, then σ-1(j) = i. The inverse undoes the original mapping.
3) Order of a Permutation
If the cycle lengths are l₁, l₂, ..., lᵣ, then ord(σ) = lcm(l₁, l₂, ..., lᵣ).
4) Inversion Count
For one-line notation [σ(1), σ(2), ..., σ(n)], inversions are pairs (i, j) with i < j and σ(i) > σ(j).
5) Parity
The sign is sgn(σ) = (-1)inv(σ). Even inversion count gives an even permutation. Odd inversion count gives an odd permutation.
6) Fixed Points and Support
A fixed point satisfies σ(i) = i. Support size equals n - fixed points.
7) Generated Subgroup
The calculator enumerates closure from ⟨A, B⟩ using compositions of A, B, and their inverses, subject to the search limit.
How to Use This Calculator
- Enter the degree n of your permutation set.
- Type permutation A and optionally permutation B.
- Use one-line or cycle notation.
- Set exponents for Ak and Bm.
- Choose a subgroup search limit for closure checks.
- Click Calculate Permutation Group.
- Review orders, parity, inversions, fixed points, support, orbits, and compositions.
- Use CSV or PDF export to save your report.
FAQs
1) What notation can I enter?
You can enter one-line notation such as 2 3 1 5 4 or cycle notation such as (1 2 3)(4 5). Both forms are accepted for each permutation field.
2) What does A ∘ B mean here?
This calculator uses the standard right-to-left rule. It applies B first and then A. That convention is shown above the results table.
3) How is the order calculated?
The order is the least common multiple of all nontrivial cycle lengths. For example, a permutation with cycle lengths 3 and 2 has order 6.
4) How is parity found?
Parity comes from the inversion count of the one-line form. An even number of inversions gives an even permutation. An odd number gives an odd permutation.
5) What is the subgroup search limit for?
The limit prevents excessively large closure searches. If the generated subgroup reaches the limit, the size shown becomes a lower bound instead of an exact total.
6) Can I leave permutation B empty?
Yes. A blank B field is treated as the identity permutation. That is useful when you only want a detailed analysis of one permutation.
7) What does the graph show?
The Plotly graph compares mapping outputs for A, B, A ∘ B, and B ∘ A across all positions from 1 to n. It helps visualize permutation action patterns.
8) Why might two permutations be conjugate?
Inside the symmetric group on the same degree, two permutations are conjugate exactly when they have the same cycle structure, including fixed points.