Note that in a general case, the group of order p^n is not necessarily abelian. However, by the given inductive hypotheis, we show that the above group G of order p^n is abelian. We know that Z(G) is non-trivial for every nontrivial finite p-group and is a subgroup of G. Now, by inductive hypothesis, G/Z(G) is cyclic. Thus G is abelian.
By the classification theorem of finitely generated abelian groups and inducitive hypothesis, or . The former is contradictory to the inductive hypothesis because there are more than one subgroup of order p in G. Thus and G is cyclic.