Enter two permutations σ and τ in cycle notation.
Enter σ and τ with the same cycle structure.
Compute g such that σ = g⁻¹ τ g (and hence show that σ and τ are conjugate).
Click to compute (a b) ∘ σ and see how the cycle structure changes.
Enter a permutation (2-row or cycle notation)
Enter a permutation (2-row or cycle notation)
The symmetric group Sn is the group of all permutations of n elements, with function composition as the group operation. It has n! elements.
A permutation can be written as a product of disjoint cycles. For example, (1 2 3)
means 1 → 2 → 3 → 1. The notation (1 2 3)(4 5) represents a permutation where
1, 2, 3 cycle among themselves and 4, 5 swap.
Order: The smallest positive integer k such that σk = identity. It equals the LCM of the cycle lengths.
• |Sn| = n!
• Sn is non-abelian for n ≥ 3