Permutation Group Calculator | Cycle Notation, Composition & Order
Work with permutations in cycle notation. Compose two permutations, compute the inverse, raise a permutation to any power, and find its order. Displays each permutation in two-line notation, full cycle decomposition, sign (even or odd), and supports the symmetric group Sₙ for n up to 8.
Elements: 1 to 4
What Is the Permutation Group Calculator | Cycle Notation, Composition & Order?
Permutations in the symmetric group Sₙ are bijections on {1,…,n}. Every permutation decomposes uniquely into disjoint cycles. Composition σ∘τ applies τ first then σ (right-to-left). The order is the LCM of all cycle lengths — the smallest k such that σᵏ = identity. The inverse σ⁻¹ reverses every cycle: (a b c)⁻¹ = (c b a). The sign (signature) is +1 if the permutation is a product of an even number of transpositions (even permutation), −1 otherwise. The sign equals (−1) raised to the number of even-length cycles.
Formula
σ∘τ(i) = σ(τ(i)) | ord(σ) = min k>0: σᵏ=e | sgn(σ) = (−1)^(even-length cycle count)
How to Use
- 1
Set n to the size of the symmetric group (elements are 1 to n, maximum n = 8)
- 2
Enter σ in cycle notation — e.g. (1 2 3)(4 5) uses spaces between elements in each cycle
- 3
Enter τ the same way, or use e (or leave blank) for the identity
- 4
Set k to compute σ^k (negative k computes powers of σ⁻¹)
- 5
Click a preset to load S₃ rotations or S₄ examples
- 6
Click "Compute" to see composition, inverse, power, order, sign, and whether σ and τ commute
Set n (size of Sₙ), enter σ and τ in cycle notation, choose k for σ^k, then click Compute.
Example Calculation
S₄: σ=(1 2 3), τ=(2 3 4). σ∘τ: trace each element — 1→τ(1)=1→σ(1)=2, 2→τ(2)=3→σ(3)=1, 3→τ(3)=4→σ(4)=4, 4→τ(4)=2→σ(2)=3. Result: (1 2)(3 4). τ∘σ=(1 3 4 2) ≠ σ∘τ so they do not commute. Order(σ)=3. Sign(σ)=+1 (one 3-cycle = even).
Understanding Permutation Group | Cycle Notation, Composition & Order
Symmetric Group Sₙ — Key Facts
| Group | Order |Sₙ| | Even permutations |Aₙ| | Is abelian? | Smallest non-trivial normal subgroup |
|---|---|---|---|---|
| S₂ | 2 | 1 | Yes | Trivial |
| S₃ | 6 | 3 | No | A₃ ≅ ℤ₃ (rotations of triangle) |
| S₄ | 24 | 12 | No | A₄ (no subgroup of order 6) |
| S₅ | 120 | 60 | No | A₅ (simple group — no normal subgroups) |
| S₆ | 720 | 360 | No | A₆ |
| S₇ | 5040 | 2520 | No | A₇ |
| S₈ | 40320 | 20160 | No | A₈ |
Order of Permutations by Cycle Structure
The order of a permutation equals the LCM of its cycle lengths.
| Cycle type in S₆ | Example | Order (LCM) | Sign |
|---|---|---|---|
| (1)(1)(1)(1)(1)(1) | identity e | 1 | +1 (even) |
| (2)(1)(1)(1)(1) | (1 2) | 2 | −1 (odd) |
| (2)(2)(1)(1) | (1 2)(3 4) | 2 | +1 (even) |
| (3)(1)(1)(1) | (1 2 3) | 3 | +1 (even) |
| (2)(2)(2) | (1 2)(3 4)(5 6) | 2 | −1 (odd) |
| (4)(1)(1) | (1 2 3 4) | 4 | −1 (odd) |
| (3)(2)(1) | (1 2 3)(4 5) | 6 | −1 (odd) |
| (6) | (1 2 3 4 5 6) | 6 | −1 (odd) |
Where Permutation Groups Appear
- ›Rubik's Cube — the cube group is a subgroup of S₄₈ with 4.3 × 10¹⁹ elements.
- ›Cryptography — permutation groups underpin DES encryption S-boxes and block cipher design.
- ›Galois theory — the Galois group of a polynomial is a permutation group; solvability by radicals requires a solvable Galois group.
- ›Particle physics — exchange symmetry of identical particles is described by representations of Sₙ.
- ›Card shuffling — a riffle shuffle is a permutation; mathematicians proved 7 riffle shuffles randomize a 52-card deck.
Frequently Asked Questions
What is cycle notation?
Cycle notation writes a permutation as a product of cycles. (1 2 3) means 1→2, 2→3, 3→1. Fixed points are omitted. The identity has no non-trivial cycles and is written e. Disjoint cycles commute with each other.
How is the order computed?
The order of a permutation equals the LCM of the lengths of its disjoint cycles. (1 2)(3 4 5) has cycles of length 2 and 3, so order = lcm(2,3) = 6.
What is an even permutation?
A permutation is even if it is a product of an even number of transpositions (2-cycles). Sign = +1. Odd permutations have sign = −1. The set of all even permutations in Sₙ forms the alternating group Aₙ, which has order n!/2.
Why does composition apply right-to-left?
By mathematical convention, σ∘τ means "apply τ first, then σ" — the same left-to-right reading as function composition f(g(x)). Some textbooks use left-to-right; always check the convention in your source.
What is the connection to the Rubik's Cube?
Each move of the Rubik's Cube is a permutation of the 48 facelets (or 20 pieces). Solving the cube is equivalent to finding the inverse permutation. The cube group has 4.3 × 10¹⁹ elements — a subgroup of S₄₈.
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