1. The problem statement, all variables and given/known data Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic. 2. Relevant equations None 3. The attempt at a solution " => " If the group, call it G, is a union of proper subgroups, then, for every subgroup, there is at least one element of G that is not in that particular subgroup. But then we know that none of the subgroups can represent all the elements of G. Therefore, G is not cyclic. " <= " If the group is not cyclic, then no element a in G generates G. That means that G is the union of all the <a> subgroups for all a in G. Is this correct?