Calculator
Enter a primary cycle notation permutation. Add a second permutation if you also want a composition.
Example data table
Example using n = 6 and σ = (1 2 3)(4 5).
| Element x | σ(x) | σ⁻¹(x) | Cycle block | Status |
|---|---|---|---|---|
| 1 | 2 | 3 | (1 2 3) | Moved |
| 2 | 3 | 1 | (1 2 3) | Moved |
| 3 | 1 | 2 | (1 2 3) | Moved |
| 4 | 5 | 5 | (4 5) | Moved |
| 5 | 4 | 4 | (4 5) | Moved |
| 6 | 6 | 6 | () | Fixed |
Formula used
For a cycle (a₁ a₂ ... ak), the permutation sends a₁ → a₂, a₂ → a₃, and ak → a₁.
Composition follows standard function order. The expression (σ ∘ τ)(x) means apply τ first, then apply σ.
The inverse reverses each cycle direction. So (1 2 3) becomes (1 3 2).
The order of a permutation is the least common multiple of the lengths of its disjoint cycles.
Parity comes from the number of transpositions. A cycle of length m uses m - 1 transpositions.
How to use this calculator
- Enter the set size n. The calculator assumes elements are 1 through n.
- Type the primary permutation in cycle notation, such as (1 2 3)(4 5).
- Optionally enter a second permutation if you want a composition result.
- Choose the composition order. This matters whenever a second permutation is supplied.
- Enter an exponent k to compute σ^k. Negative powers use the inverse automatically.
- Pick one element to inspect. The calculator shows how that element moves under several related permutations.
- Press Calculate permutation. The results will appear above the form.
- Use the CSV and PDF buttons to save the generated report and mapping table.
Frequently asked questions
1. What is cycle notation?
Cycle notation groups elements that move in a loop. The cycle (1 2 3) means 1 goes to 2, 2 goes to 3, and 3 returns to 1.
2. What does an omitted number mean?
If an element from 1 through n does not appear in the written cycles, it is treated as a fixed point. The calculator lists those values separately.
3. Why does composition order matter?
Permutation composition is not usually commutative. Applying τ then σ can produce a different result than applying σ then τ, even when both use the same set.
4. How is the order of a permutation found?
Convert the permutation into disjoint cycles, then take the least common multiple of the cycle lengths. That value is the smallest positive exponent returning the identity.
5. How do you decide whether a permutation is even or odd?
A permutation is even when it can be written using an even number of transpositions. It is odd when the total number of required transpositions is odd.
6. Can I use negative exponents?
Yes. The calculator interprets negative powers with the inverse permutation. For example, σ^-2 means apply the inverse twice, or square σ⁻¹.
7. What format should I type inside each cycle?
Use integers separated by spaces or commas, wrapped in parentheses. Examples include (1 2 3), (4,5), or (1 2)(3 4 5).
8. What does the graph show?
The graph plots each element on the horizontal axis and its image on the vertical axis. It provides a fast visual summary of movement across the set.