Permutation Groups Preview

Combinatorics · Axiom Academy

LESSON Permutation Groups Preview Discovering the algebraic structure behind permutations and the symmetric group A group is a set with an operation satisfying four properties: closure, associativity, identity, and inverses. Permutations of n elements form a group under composition . When we compose two permutations σ and τ, written σ ∘ τ, we apply τ first, then σ. This creates a new permutation that's also in the group. 2. Cycle Notation for Permutations Instead of listing where each element goes, we can use cycle notation to show the cyclic pattern of movements. The cycle (1 2 3) means: 1→2, 2→3, 3→1. Disjoint cycles (those with no elements in common) can be written in any order and commute with each other. This notation reveals the underlying structure of permutations. To compose permutations in cycle notation, we apply them from right to left (following function composition convention). Track where each element goes through both permutations. Following element 1: τ fixes it, σ sends it to 2. Following element 2: τ sends 2→3, σ sends 3→1. This process reveals the composition's cycle structure. The symmetric group S n contains exactly n! elements—all possible permutations of n objects. For example, S 3 has 3! = 6 permutations, and S 4 has 4! = 24 permutations. 5. Connection to Abstract Algebra The symmetric group S n is fundamental to abstract algebra. It's the most "symmetric" group in that every finite group is a subgroup of some S n (Cayley's theorem).

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