Enter Permutation Data
Example Data Table
| Permutation Input | Notation | Degree | Parity | In An? | Order |
|---|---|---|---|---|---|
| 2 3 1 4 | One-line | 4 | Even | Yes | 3 |
| 2 1 3 4 | One-line | 4 | Odd | No | 2 |
| (1 2 3)(4 5) | Cycle | 5 | Odd | No | 6 |
| (1 2 3)(4 5 6) | Cycle | 6 | Even | Yes | 3 |
Formula Used
Alternating group test: A permutation belongs to An exactly when it is even.
Sign from inversions: sgn(σ) = (-1)^{I(σ)}, where I(σ) is the inversion count.
Membership rule: if I(σ) is even, then sgn(σ) = +1 and σ ∈ A_n.
Order of a permutation: the permutation order is the least common multiple of all cycle lengths.
Order of the alternating group: |A_n| = n! / 2 for n ≥ 2, while |A_1| = 1.
This calculator counts inversions from the normalized one-line form, computes the sign, converts the permutation into cycle form, then reports whether the permutation is inside the alternating group.
How to Use This Calculator
- Enter a permutation in one-line notation or disjoint cycle notation.
- Provide the degree n if you want a larger ambient symmetric group.
- Select the notation style, or keep auto-detect enabled.
- Click Check Alternating Membership to analyze the permutation.
- Read the summary above the form for parity, sign, order, support, and membership.
- Use the export buttons to download the result as CSV or PDF.
Frequently Asked Questions
1. What does the calculator decide?
It decides whether your permutation is an even permutation, which means it belongs to the alternating group An. It also reports sign, inversions, cycle form, order, support, and fixed points.
2. What notation can I enter?
You can enter one-line notation like 2 3 1 4 or cycle notation like (1 2 3)(4 5). The auto option detects the format from your input.
3. Why does parity determine membership?
The alternating group contains exactly the even permutations in the symmetric group. If a permutation has sign +1, it is inside An; if sign is -1, it is outside.
4. What is an inversion?
An inversion is a pair of positions i and j with i < j but σ(i) > σ(j). Even inversion counts give even permutations, while odd counts give odd permutations.
5. How is permutation order computed?
The order comes from the least common multiple of the cycle lengths. For example, cycle lengths 3 and 2 give permutation order 6.
6. Why can degree be larger than the moved elements?
A permutation may act inside a larger symmetric group by leaving extra elements fixed. Setting a larger degree lets the calculator include those fixed points in the analysis.
7. Does the calculator validate the permutation?
Yes. It checks that one-line notation contains every integer from 1 through n exactly once, and that cycle notation uses valid disjoint cycles without repeated elements.
8. What do the CSV and PDF downloads contain?
The downloads contain the computed result summary, including membership, sign, order, inversion count, cycle form, fixed points, and support, so you can keep or share your analysis.