36th Putnam 1975 Problem B1 Let G be the group { (m, n) : m, n are integers } with the operation (a, b) + (c, d) = (a + c, b + d). Let H be the smallest subgroup containing (3, 8), (4, -1) and (5, 4). Let H ab be the smallest subgroup containing (0, a) and (1, b). Find a > 0 such that H ab = H. Solution
: 7. (3,8) = 3 (1,5) - (0,7), (4,-1) = 4 (1,5) - 3 (0,7), (5,4) = 5 (1,5) - 3 (0,7). (1,5) = (5,4) - (4,-1), (0,7) = - 4 (3,8) - 7 (4,-1) + 8 (5,4). This shows that { (3,8), (4,-1), (5,4) } and { (1,5), (0,7) } generate the same subgroups. 36th Putnam 1975 © John Scholes jscholes@kalva.demon.co.uk 27 Jan 2001