Hi any help with this is greatly appreciated!

1) Consider the (multiplicative) quaternion group Q = {1, i, j, k} defined by the following relation: .i^2 = j^2 = k^2 = 1, ij = k, jk = i, ki = j, ji = k, kj = i, ik = j, x1 = 1x = x & x(1) = (1)x = x. Find all subgroups of Q, and give the lattice of subgroups. You should prove that you have all the subgroups.

2)Let G be a group that has no subgroups at all, except for the two trivial ones: {e} and G itself. Assume that G contains more than just the identity, i.e. G does not equal {e} (a) Show that if a G, that G = . (b) Show that if a G, then the order of a must be finite. (hint: consider and ) (c) Show that if a G, then the order of a must be prime. (hint: consider and . . . ) (d) Conclude that G must be a cyclic group of prime order.