[SOLVED] group theorem problems 1. The problem statement, all variables and given/known data 1) suppose H is a finite subset of a group G. Prove that H is a subgroup of G if and only if xy belongs to H whenever x and y belong to H 2) show that Q is not cyclic, (where Q is the group of rationals with addition.) 3. The attempt at a solution 1) if H is a subgroup, then if x,y are in H, so is by hypothesis xy, because H is a subgroup and therefore closed. Now conversely, if xy is an element of H when x,y are in H, then x*x is in H Let n be the number of elements in H, then x^n is in H Now because H is finite, there has to be an element such that x^n+1=x^m, with n+1>m. Otherwise there would be n+1 distinct elements in a set with n elements. I didn't get any further then this. I suppose if for elements of H a*b=a*c would imply that b=c, then x=x^n+1-m and so e:=x^n-m. Then I can find and inverse too without many problems, and because H is already know to have an associative operation, and H is closed by hypothesis, H is a subgroup. But is this true? 2) Suppose there is an element x such that <x>=Q. Then x/2 is too in Q, and is not in <x>. But I don't find this a very convincing proof by contradiction.