Conjugacy Classes in Sₙ

Abstract Algebra · Axiom Academy

LESSON Conjugacy Classes in Sₙ Discovering the beautiful connection between conjugation and cycle types in the symmetric group Geometrically, conjugation by τ "relabels" the elements being permuted. If σ maps i → j, then τστ⁻¹ maps τ(i) → τ(j). It's like performing σ in a different coordinate system! This means conjugacy classes in Sₙ are completely determined by cycle structure! For example, all 3-cycles are conjugate to each other, and all permutations with cycle type (2,2,1) are conjugate to each other. 3. Why Does Conjugation Preserve Cycle Type? Let's see why conjugation preserves cycle structure through a concrete example. Consider σ = (1 2 3) and τ = (1 4). We'll compute τστ⁻¹ step by step. 4. Counting Conjugacy Classes via Partitions Since conjugacy classes correspond to cycle types, and cycle types correspond to partitions of n, the number of conjugacy classes in Sₙ equals the number of partitions of n, denoted p(n). Each partition λ = (λ₁, λ₂, ..., λₖ) of n gives a conjugacy class The size of the conjugacy class for partition λ is: n! / (∏ᵢ λᵢ · ∏ⱼ mⱼ!) where mⱼ is the number of parts of size j in λ For S₅: p(5) = 7 conjugacy classes

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