Analyzer Inputs
Example Data Table
Use these sample inputs to verify your setup and compare outputs.
| n | Permutation (cycle notation) | Cycle type | Order | Expected parity |
|---|---|---|---|---|
| 5 | (1 3 5)(2 4) | 3-2 | 6 | Odd |
| 6 | (1 2 3 4)(5 6) | 4-2 | 4 | Even |
| 7 | (1 7)(2 6)(3 5) | 2-2-2-1 | 2 | Odd |
Formulas Used
- Group order: |Sn| = n!.
- Parity via inversions: inv(p) = #{(i<j) : p(i)>p(j)}; even if inv(p) is even.
- Sign: sgn(p) = (−1)inv(p).
- Disjoint cycles: p decomposes uniquely into disjoint cycles.
- Permutation order: ord(p) = lcm of all cycle lengths.
- Cayley length: ℓ(p) = n − c(p), where c(p) counts cycles including fixed points.
- Centralizer size: |C(p)| = ∏ imi · (mi!), with mi cycles of length i.
- Conjugacy class size: |Cl(p)| = n! / |C(p)|.
How to Use This Calculator
- Choose n, which defines the symmetric group Sn.
- Select the input format or leave it on auto-detect.
- Enter permutation A (and B for composition or commutator).
- If using A^k, enter an integer k (negative allowed).
- Press Submit to view results above the form.
- Use Download CSV or Download PDF to export.
For one-line notation, provide exactly n numbers. For cycle notation, use disjoint cycles like (1 3 5)(2 4).
Group size and limits
This analyzer works in S_n with n from 1 to 60, balancing rich algebra output with readable tables. The group order is computed as n!, shown as an integer to support large n. The tool also reports counts of even and odd permutations, each equal to n!/2 for n ≥ 2, helping you sanity-check parity results. It also displays a practical generator count n-1, matching adjacent transpositions.
Permutation entry and operations
Enter permutations in cycle notation like (1 3 5)(2 4) or one-line notation like 2 1 3 4 5. Choose an operation to analyze A, compose A ∘ B, compose B ∘ A, invert A, raise A to k, or compute the commutator [A,B]. Composition follows (p ∘ q)(i)=p(q(i)), so order of application is explicit. Blank input is treated as the identity.
Cycle structure and invariants
After submission, the calculator decomposes the result into disjoint cycles, lists the cycle type, and computes the permutation order as the least common multiple of cycle lengths. Parity is derived from the inversion count inv(p)=#{(i<j): p(i)>p(j)}, with sign (−1)inv(p). A transposition decomposition is produced, and the minimal transposition count is reported as n - c(p), where c(p) includes fixed points. The mapping table i→p(i) supports quick verification.
Conjugacy, centralizers, and classes
Elements of S_n are conjugate exactly when they share the same cycle type, so the cycle multiplicities m_i drive class analytics. The centralizer size is |C(p)|=∏ im_i·(m_i!), and the conjugacy class size is n!/|C(p)|. The total number of conjugacy classes equals the partition count p(n), linking computations to integer partitions and representation theory. These values connect computations to orbit-stabilizer reasoning and explain why some cycle types appear far more frequently than others.
Exports and practical workflows
Use CSV export to capture key metrics for spreadsheets, lectures, or automated checking. The PDF export produces a text report that includes the group summary, cycle notation, parity, and a mapping sample, and share outputs with peers. Results are stored for download until you run a new analysis. For assignments, start with the example table, reproduce the outputs, then vary k or swap composition order to observe invariants and non-commutativity.
FAQs
What is Sn in this calculator?
Sn is the set of all permutations of {1,…,n} under composition. Its size is n!, which the calculator displays as the group order.
Why is n capped at 60?
Factorials and class sizes grow extremely fast. The cap keeps computations stable and keeps tables readable, while still supporting meaningful group exploration and verification.
How is permutation parity computed?
Parity is computed from the inversion count inv(p)=#{(i<j): p(i)>p(j)}. Even inversions give an even permutation; odd inversions give an odd permutation and sign −1.
What does the “order” of a permutation mean?
The order is the smallest positive m such that pm is the identity. The calculator computes it as the LCM of the disjoint cycle lengths.
What do centralizer and class size tell me?
The centralizer counts permutations that commute with p. The conjugacy class size counts permutations with the same cycle type, computed as n!/|C(p)|.
What is the commutator [A,B] used for?
[A,B]=A−1B−1AB measures non‑commutativity. If it equals the identity, then A and B commute; otherwise it quantifies how far they fail to commute.