Assume G is a finite abelian p-group. Since G is finite, there exists such that where . This is a maximal element of G. Now if G = <a> = then we are done. Assume G <a> . Then such that . Now since G is a group, closed under operation, so has power , Now abelian groups has a property that for . So using this property, . If then it means is the inverse of , which would mean, since <a> is a subgroup, it contains its own inverses, so which would mean which we said cannot be, so that means both and since , this means and is also a maximal element. Now you can see how this would proceed further, take an element not in or and u would come to the same conclusion about the new element and thus u can see that since G is finite, you get are maximal elements which when intersected give u {e}, and which . Since all subgroups of an abelian group G, are normal in G. then indeed G =