Chapter 11 - Group Theory and Sylow Theorems Sylow Theorems • Let |G| = pk · m, where p is a prime, p ∤ m. • The Sylow theorems guarantee: 1. Existence: G contains at least one Sylow p-subgroup of order pk . 2. Conjugacy: All Sylow p-subgroups of G are conjugate. 3. Counting: The number of Sylow p-subgroups, np , satisfies: – np ≡ 1 (mod p), – np | m. Group Actions • Groups act on sets via permutations. • Definitions: – Orbit: For x ∈ X, the set {g · x | g ∈ G} is called the orbit of x. – Stabilizer: The set of elements in G that fix x is called the stabilizer of x, denoted Gx . • Orbit-Stabilizer Theorem: |G| = |Orbit of x| · |Gx |. 1 Applications of Sylow Theorems • Normality of Sylow Subgroups: – If np = 1, then the Sylow p-subgroup is unique and normal. • Determining Group Structures: – Example: For |G| = 12: ∗ Sylow 2-subgroups: n2 ≡ 1 (mod 2), n2 | 3, so n2 = 1 or 3. ∗ Sylow 3-subgroups: n3 ≡ 1 (mod 3), n3 | 4, so n3 = 1 or 4. Diagrams and Examples Sylow Subgroup Conjugacy Diagram Conjugation by g Sylow p-subgroup P1 Sylow p-subgroup P2 Conjugation by g −1 Subgroup Lattice for A4 A4 V4 C3 {e} Examples Sylow Subgroups of S3 • Order of S3 : |S3 | = 6 = 21 · 3. 2 • Sylow 2-subgroups: – n2 ≡ 1 (mod 2), n2 | 3, so n2 = 1 or 3. • Sylow 3-subgroups: – n3 ≡ 1 (mod 3), n3 | 2, so n3 = 1 or 2. • If n2 = 1, the Sylow 2-subgroup is normal. 3