1. consider the group Q8 = {1,-1,i,-i,j,-j,k,-k} where i^2 = j^2 = k^2 = -1, ij = k, jk = i, and ki = j. what can you say about the proper subgroups of Q8?

2. this is a first-rate question with which to test your knowledge of groups. here is an idea: consider any non-identity element g1, you can always form <g1>.

now if <g1> was infinite, it would be isomorphic to Z, which has an infinite number of proper subgroups, which would then be an infinite number of proper subgroups of G.

so we may safely conclude that <g1> is finite. if G = <g1>, we are done. if not, then there is some g2 not in <g1>. consider <g2>.

now, prove that the number of subgroups we find in this way must be finite. why does this suffice to show that G is finite?