Practice Problems 1. 11.2 Q1 • Let |G| = 56. Determine the number of Sylow 2-subgroups and Sylow 7-subgroups in G. Are any of these subgroups normal? Solution: • The order of G is |G| = 56 = 23 · 7. By Sylow’s theorems: – The number of Sylow 2-subgroups, n2 , satisfies n2 ≡ 1 (mod 2) and n2 | 7. Thus, n2 = 1 or 7. – The number of Sylow 7-subgroups, n7 , satisfies n7 ≡ 1 (mod 7) and n7 | 8. Thus, n7 = 1 or 8. • If n2 = 1, the Sylow 2-subgroup is unique and normal. • If n7 = 1, the Sylow 7-subgroup is unique and normal. • Without additional structure, both cases are possible. 2. 11.2 Q2 • Prove that all Sylow p-subgroups of a group G are conjugate. Solution: • Let P1 and P2 be two Sylow p-subgroups of G. • Consider the action of G on the set of Sylow p-subgroups by conjugation. • This induces a homomorphism ϕ : G → Sym(S), where S is the set of Sylow p-subgroups. • By Sylow’s theorem, the stabilizer of P1 is P1 itself. • The orbit-stabilizer theorem implies that |S| = np , where np ≡ 1 (mod p). • Hence, all Sylow p-subgroups are conjugate. 3. 11.5 Q2 • Determine the Sylow subgroups of A4 and whether any are normal. 1 Solution: • The order of A4 is |A4 | = 12 = 22 · 3. • For Sylow 2-subgroups: – n2 ≡ 1 (mod 2) and n2 | 3. Thus, n2 = 1 or 3. • For Sylow 3-subgroups: – n3 ≡ 1 (mod 3) and n3 | 4. Thus, n3 = 1 or 4. • If n2 = 1, the Sylow 2-subgroup is normal. If n3 = 1, the Sylow 3-subgroup is normal. 4. 11.5 Q3 • Verify that the matrices A and B in GL2 (R) 0 1 0 A= , B= −1 0 −1 given by: 1 1 • Form a subgroup under matrix multiplication. Solution: • 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 • Verify closure under multiplication and identity. • Conclude that {I2 , A, A2 , B, B 2 , B 3 } forms a subgroup. 5. 11.5 Q4 • Draw and analyze the subgroup lattice of Q8 . Solution: 2 • The subgroup lattice includes: – {e}. – Center Z(Q8 ) = {e, −1}. – Four cyclic subgroups of order 4. • Use TikZ for visualization: Q8 Z(Q8 ) {e} 3