- 5 Consider The Dihedral Group D6 Whose Subgroups Are As Follows Ro R1 82 F3 14 P5 M1 M2 M3 M4 M5 Mo Who 1 (429.88 KiB) Viewed 16 times
= 5. Consider the dihedral group D6 whose subgroups are as follows. {ro,r1, 82, f3, 14, P5, M1, M2, M3, M4, M5, mo}, whose identity is ro and QUSTAO Copy (saboro (m3), (m. Copyri? O 122 2022 able SK UY 2 • Order 1: {ro} • Order 2: (r3), • Order 3: (r2) , 13, m2, m5}, {ro,r3, m3, mo} • Order 6: (r1), {ro, 12, 14, mì, m3, m5}, {ro, r2, 14, m2, m4, m6} Order 12: D6 1 Order 4: {ro, r3, m1, m4}, {re le U copyright Mon opyright Monas sessable task. Copyright Monash Us cassable task. Copyright Monash Unie Copyright Monash Univa Copyright Monash Uni ensable SITY versity (a) Consider the function f :D6 + D3 definere SZALL wersity 20 2. Sity 20 ASSE SSESS o ssa. Ob oht MORE right BESC Ole task sessable task, Copyright Monash for 0 si55. pyright Monash f(ri) = l (i mod 3 f(m;) = (j mod sopyright Monash Copyright Mona opyright Mo syright right = for 1 < i < 6, e able task, Copyright Monash Univer ble, Jnivele Universy ash Universiti onash Diversity Monk (6) where we write m3 = mo in D3. Use the fact that at f is a homomorphism to prove that (r3 ) is a normal subgroup of Do and that D6/(r3 ) = D3. (You do not have to prove that f is a homomorphism.) Draw the lattice of subgroups of D6 and circle those subgroups that contain ( 13 ). Draw the lattice of subgroups of D3. (Hint. There are six subgroups of D3.) (d) What does the correspondence theorem tell you about the two lattices that you have drawn? ору Task task