Question

(a) List all the generators of < 5 > in Z6o. (b) List all of the left cosets of < 10 > in the subgroup < 2 > of Z60.

Answer 1

Date: 11 (25) = 0 of H. = 45 a = Zoo, HSG e, H= <s> M-Zos,10,15,20, 25, 30, 35,40,45,50, 55} Now 0(4)=12 generators of t have order 12 : elements of order 12 are generators row a 5k)= 0(5) R=1,2 --- ged (K, 12) do)= 1 as o is the identity element 015) = 12 as H=<5> Olio): 0(245) 12 = 6 god (312) 0(15)= 0(3x5)= 12 ged (312) 0(20) 04x5)= 12 gedly,12) 12 2 4 (2 3 3 12 4 ✓ 12 12 =12 1 12 2 0(25)= 0(5x5)= gd (5,12) 0(30)= 0 (6x5)= ged (6,12) 0(38)= 0/7x8) = gid(7,12) 0(40) 08xs) = 12_ ged (8,12) 12 12 12 - 1 12= 3 4

2 0 (45) 0(x)= 12 12 = 4 붕 gd(9,12) 050)Ol10x5) 12 12-6 ged(10, 12) 0(56)= olix)= 123 12 = 12 V ged(11,12) 1 : elements of order 12 in <57 { 5, 25, 35, 55} generators of <57 in Zoo are 15, 25, 35, 55] 6 G = 7o, 1 = <27, K=<10> now H= 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 582 K K={0, 10, 20, 30, 40, 503 _-4x+x) REM} th-{0+X,9+X, 4+R, 6+K, 8+X, 10K, 12+K, R 14+ K, 76+8, 18+K, 2015, 221K, 24+K, 2613, 28+ 30 +K_3278, 34+ K, 36+%, 38 +K_40tk, 42+K, 4416,46+ K, 48 +K, 501K, 527K, 54+k, 36tR 58 KB

now 20 +K = K x 26eK 2otK= К А Я ОК 30 + К= КщЗоєК ЧО+KE K as 40EK 50+ K = as soek :. - 4K, 2-к, 4+K, +6 +k, 8+к, 12+к, 14+K, 6+к, 18+К, 22tK 24+K, 26ЕК, 24+K ,32+K, 31tK, 36 tK, 33 tK , 42+к, чч+К, Ч6+к, 48+К, S2+, Sy+K, LtK, S8+k } H = {27, К- Ко>