Calculated Results
Your symmetric group analysis will appear here.
Permutation Mapping Table
| i | p(i) | q(i) | (p ∘ q)(i) | pk(i) | Inverse p-1(i) |
|---|
Plotly Graph
The chart compares the identity map with the selected permutation and optional composition.
Calculator Inputs
Enter a degree n and one-line notation for permutation p. Add permutation q if you want composition. Use the exponent k to compute pk.
Example Data Table
These sample rows show the kind of outputs the calculator produces for group size, parity, composition, and powers.
| n | Permutation p | Permutation q | k | Order of p | Parity | p ∘ q | pk |
|---|---|---|---|---|---|---|---|
| 5 | 2 3 1 5 4 | 3 1 2 5 4 | 3 | 6 | Odd | 1 2 3 4 5 | 1 2 3 5 4 |
| 4 | 2 1 4 3 | 2 1 3 4 | 2 | 2 | Even | 1 2 4 3 | 1 2 3 4 |
| 6 | 2 3 4 5 6 1 | 1 3 2 4 6 5 | 4 | 6 | Odd | 2 4 3 5 1 6 | 5 6 1 2 3 4 |
Formula Used
The calculator combines standard symmetric-group formulas with permutation algorithms based on one-line notation.
|Sₙ| = n!
This counts all bijections from {1,2,...,n} to itself.
(p ∘ q)(i) = p(q(i))
The permutation q acts first. Then the result passes through p.
inv(p) = #{(i,j) : i < j and p(i) > p(j)}, sign(p) = (-1)^{inv(p)}
Even inversions give an even permutation. Odd inversions give an odd permutation.
order(p) = lcm(c₁, c₂, ..., cᵣ)
If p splits into disjoint cycles of lengths c₁, c₂, ..., cᵣ, the order is their least common multiple.
p^{-1}(p(i)) = i, p^0 = identity, p^{-k} = (p^{-1})^k
The calculator supports positive, zero, and negative exponents.
minimum transpositions = n - c
Here c is the number of cycles when fixed points are included as 1-cycles.
How to Use This Calculator
- Enter the degree n of the symmetric group Sₙ.
- Type permutation p in one-line notation using spaces or commas.
- Optionally enter permutation q if you want a composition result.
- Set exponent k to evaluate pk, including negative powers.
- Click Analyze Symmetric Group to place the full result above the form.
- Review cycle notation, parity, inversions, order, inverse, composition, and graph.
- Use the CSV or PDF buttons to export the current report.
Frequently Asked Questions
1) What does this calculator analyze?
It studies one permutation in Sₙ and can also compare a second permutation. You get cycle notation, parity, sign, inverses, powers, order, compositions, and mapping tables.
2) How should I enter a permutation?
Use one-line notation with spaces or commas, such as 2 3 1 5 4. The entries must contain each number from 1 through n exactly once.
3) What does p ∘ q mean here?
This calculator uses p ∘ q(i) = p(q(i)). So q acts first, then p. The displayed composition follows that standard function-composition convention.
4) How is parity determined?
Parity comes from the inversion count. An even number of inversions gives an even permutation. An odd number of inversions gives an odd permutation.
5) How is the order of a permutation found?
Write the permutation as disjoint cycles, then take the least common multiple of those cycle lengths. That LCM is the permutation order.
6) Can I compute negative powers?
Yes. Negative exponents are applied by first finding the inverse permutation, then raising that inverse to the corresponding positive power.
7) Why are fixed points useful?
Fixed points show which elements remain unchanged. They help identify support size, cycle structure, and whether a permutation is close to the identity.
8) What are the export buttons for?
CSV export gives a clean data file for spreadsheets. PDF export creates a compact report containing the main metrics and mapping table.