Assignment: 1. Given the generators and relations 3a + 6 b + 12 c = 0 8a + 2 b + 6 c = 0 2a + 3 b + 15 c = 0 Decompose the group into cyclic summands and give the generators of each cyclic summand. a b c 3 6 12 8 2 6 2 3 15 a b c 2 3 15 3 6 12 8 2 6 a b c 2 3 15 1 3 -3 0 -10 -54 a b c 1 3 -3 2 3 15 0 -10 -54 a b c 1 3 -3 0 -3 21 0 -10 -54 add -3 column a to column b add 3 column a to column c a+3b-3c b c 1 0 0 0 -3 21 0 -10 -54 a+3b-3c b c 1 0 0 0 -3 21 0 -1 -117 a+3b-3c b c 1 0 0 0 -1 -117 0 -3 21 a+3b-3c b c 1 0 0 0 +1 +117 0 -3 21 a+3b-3c b c 1 0 0 0 +1 +117 0 0 372 add -117 column 2 to column 3 a+3b-3c b+117c c 1 0 0 0 1 0 0 0 372 Z1 x Z1 x Z372 Essentially the group is cyclic of order 372. a' = a+3b-3c = 0 b' = b+117c = 0 c' = c A = {{1,3,-3},{0,1,117},{0,0,1}} | a'| | 1 3 -3 || a | | b'| = | 0 1 117 || b | | c'| | 0 0 1 || c | | a | | 1 -3 354 || a'| | b | = | 0 1 -117 || b'| | c | | 0 0 1 || c'| 0 x 0 x Z372 a = (0, 0, 354) b = (0, 0, -117) c = (0, 0, 1) 3a + 6 b + 12 c = (0,0,3*354-6*117+12) = (0,0, 372) = (0,0,0) 8a + 2 b + 6 c = (0,0,8*354-2*117 +6) = (0,0,2604) = (0,0,0) 2a + 3 b + 15 c = (0,0,2*354-3*117+15) = (0,0, 372) = (0,0,0_ 2. Given the generators and relations 8a + 5 b + 12 c = 0 6a + 3 b + 2 c = 0 6a + 9 b + 6 c = 0 Decompose the group into cyclic summands and give the generators of each cyclic summand. a b c 8 5 12 6 3 2 6 9 6 a b c 8 5 12 6 3 (2) 6 9 6 a b c 6 3 (2) 8 5 12 6 9 6 c a b (2) 6 3 12 8 5 6 6 9 c a b (2) 6 3 0 -28 -13 0 -12 0 add -3 times column 1 to column 2 add -1 times column 1 to column 3 3a+b+c a b (2) 0 1 0 -28 -13 0 -12 0 3a+b+c a b 2 0 (1) 0 -28 -13 0 -12 0 b 3a+b+c a (1) 2 0 -13 0 -28 0 0 -12 b 3a+b+c a (1) 2 0 0 26 -28 0 0 -12 multiply -2 times column 1 and add it to column 2 6a+3b+2c 3a+b+c a (1) 0 0 0 26 -28 0 0 -12 6a+3b+2c 3a+b+c a (1) 0 0 0 26 -28 0 0 (-12) 6a+3b+2c 3a+b+c a (1) 0 0 0 0 (-12) 0 26 -28 6a+3b+2c a 3a+b+c (1) 0 0 0 (-12) 0 0 -28 26 6a+3b+2c a 3a+b+c (1) 0 0 0 (+12) 0 0 -28 26 6a+3b+2c a 3a+b+c (1) 0 0 0 (+12) 0 0 -4 26 6a+3b+2c a 3a+b+c (1) 0 0 0 (-4) 26 0 +12 0 6a+3b+2c a 3a+b+c (1) 0 0 0 (+4) -26 0 +12 0 6a+3b+2c a 3a+b+c (1) 0 0 0 (+4) -26 0 0 78 add 7 times column 2 to column 3 6a+3b+2c -20a-7b-7c 3a+b+c (1) 0 0 0 (+4) 2 0 0 78 6a+3b+2c -20a-7b-7c 3a+b+c (1) 0 0 0 +4 (2) 0 0 78 6a+3b+2c 3a+b+c -20a-7b-7c (1) 0 0 0 (2) +4 0 78 0 6a+3b+2c 3a+b+c -20a-7b-7c (1) 0 0 0 (2) +4 0 0 -156 add -2 times column 2 to column 3 6a+3b+2c -37a-13b-13c -20a-7b-7c (1) 0 0 0 (2) 0 0 0 -156 6a+3b+2c -37a-13b-13c -20a-7b-7c (1) 0 0 0 (2) 0 0 0 +156 Z1 x Z2 x Z156 | a'| | 6 3 2 || a | | b'| = |-37 -13 -13 || b | | c'| |-20 -7 -7 || c | | a | | 0 7 -13 || a'| | b | = | 1 -2 4 || b'| | c | | -1 -18 33 || c'| 3. Given the generators and relations 12a + 9 b + 2 c = 0 8a + 3 b + 4 c = 0 4a + 4 b + 6 c = 0 Decompose the group into cyclic summands and give the generators of each cyclic summand. a b c 12 9 2 8 3 4 4 4 6 c a b 2 12 9 4 8 3 6 4 4 c a b 2 12 9 0 -16 -15 0 -32 -23 add -6 column 1 to column 2 add -4 column 1 to column 3 6a+4b+c a b 2 0 1 0 -16 -15 0 -32 -23 b 6a+4b+c a 1 2 0 -15 0 -16 -23 0 -32 b 6a+4b+c a 1 2 0 0 30 -16 0 46 -32 add -2 times column 1 to column 2 12a+9b+2c 6a+4b+c a 1 0 0 0 30 -16 0 46 -32 12a+9b+2c a 6a+4b+c 1 0 0 0 -16 30 0 -32 46 add 2 times column 2 to column 3 12a+9b+2 -11a-8b-2c 6a+4b+c 1 0 0 0 -16 -2 0 -32 -18 12a+9b+2 6a+4b+c -11a-8b-2c 1 0 0 0 -2 -16 0 -18 -32 12a+9b+2 6a+4b+c -11a-8b-2c 1 0 0 0 +2 +16 0 -18 -32 12a+9b+2 -82a-60b-15c -11a-8b-2c 1 0 0 0 +2 0 0 0 112 Z1 x Z2 x Z112 |a'| | 12 9 2 | |a | |b'|=|-82 -60 -15 | |b | |c'| |-11 -8 -2 | |c | |a | | 0 2 -15| |a'| |b |=| 1 -2 16| |b'| |c | | -4 -3 18| |c'| 12a + 9 b + 2 c = (0, 0, 0) = (0,0,0) 8a + 3 b + 4 c = (0, -2, 0) = (0,0,0) 4a + 4 b + 6 c = (0,-18,112) = (0,0,0)