Finding Sylow Subgroups of S₄

Abstract Algebra · Axiom Academy

EXAMPLE Finding Sylow Subgroups of S₄ Apply Sylow theorems to systematically find all Sylow 2-subgroups and Sylow 3-subgroups of the symmetric group S₄. Excellent work! You've systematically found all Sylow subgroups of S₄. Here's what we learned: Order of S₄: |S₄| = 24 = 2³ × 3, which determines possible Sylow subgroups Sylow 2-subgroups: Have order 8, and there are exactly n₂ = 3 of them (all conjugate) Sylow 3-subgroups: Have order 3, and there are exactly n₃ = 4 of them (all conjugate) Sylow's Third Theorem: Constrains n_p by requiring n_p ≡ 1 (mod p) and n_p | m where |G| = pᵏm Conjugacy: All Sylow p-subgroups are conjugate to each other, forming a single conjugacy class Explicit construction: Sylow 2-subgroups can be realized as copies of D₄, while Sylow 3-subgroups are generated by 3-cycles This method generalizes to finding Sylow subgroups in any finite group. The Sylow theorems provide powerful counting arguments before we even construct explicit examples!

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