Chapter 11: Study Notes on Group Theory Chapter 11: Sylow Theorems and Their Applications • Sylow Theorems: Overview – Let |G| = pk m where p is a prime and p ∤ m. – Sylow Theorems guarantee: 1. Existence: G has a subgroup of order pk (a Sylow p-subgroup). 2. Conjugacy: All Sylow p-subgroups are conjugate. 3. Counting: The number of Sylow p-subgroups, np , satisfies: ∗ np ≡ 1 (mod p). ∗ np | m. • Applications of Sylow Theorems: – Proving Normality: ∗ If np = 1, the Sylow p-subgroup is unique and thus normal. – Non-simple Groups: ∗ Example: |G| = 12. ∗ Sylow 2-subgroups: n2 ≡ 1 (mod 2) and n2 | 3. Thus, n2 = 1 or 3. ∗ If n2 = 1, the Sylow 2-subgroup is normal. • Matrix Representations of Groups: – Example: Consider the matrices A and B in 0 1 0 A= , B= −1 0 −1 1 GL2 (R): 1 . 1 – Compute powers of A: −1 0 A = = −I2 . 0 −1 2 – Compute powers of B: −1 1 2 B = , −1 0 −1 0 B = = −I2 . 0 −1 3 – These matrices form a subgroup of GL2 (R) under matrix multiplication. – The group is cyclic with A and B satisfying A2 = B 3 = −I2 . • Visual Representation of Conjugacy: Conjugation by g ∈ G Sylow p-subgroup P1 Sylow p-subgroup P2 Conjugation by g −1 • Examples: – S3 : Order 6 = 21 · 3. ∗ Sylow 2-subgroup: Order 2, {(1), (12)}. ∗ Sylow 3-subgroup: Order 3, {(1), (123), (132)}. ∗ n2 = 1, so Sylow 2-subgroup is normal. – A4 : Order 12 = 22 · 3. ∗ Sylow 2-subgroups: n2 ≡ 1 (mod 2), n2 | 3. Possible n2 = 1 or 3. ∗ Sylow 3-subgroups: n3 ≡ 1 (mod 3), n3 | 4. Possible n3 = 1 or 4. ∗ Use subgroup structure to identify normal subgroups. ∗ Diagram showing the structure of A4 : 2 A4 V4 (Klein 4) C3 (Cyclic) {e} – Q8 (Quaternion Group): Order 8 = 23 . ∗ All Sylow 2-subgroups are the group itself since n2 = 1. ∗ Visualizing Q8 as a subgroup lattice: Q8 Z(Q8 ) {e} • Key Points to Remember: – Sylow Theorems are fundamental tools for analyzing group structures. – Counting the number of Sylow subgroups provides insights into group normality and simplicity. – Applications extend to proving normality, constructing examples, and identifying subgroup lattices. – Visualization of subgroup lattices aids in understanding complex group structures. 3