Construction of a surjective group homomorphism

Question

Suppose $G$ is a noncyclic p-group which have order $p^n$ where $n\ge 2$. Is there a surjective group homomorphism from $G$ to $\mathbb{Z}_p\times \mathbb{Z}_p$? I know for $n=2$, this is true. Thank you.

Answer 1

It is clear that this holds if $G$ is abelian. If $G$ is not abelian, then it is certainly nilpotent. Say it is nilpotent of class exactly $k$, then letting $\{e\}=Z_0(G)\lt Z_1(G)\lt\cdots\lt Z_{k-1}(G)\lt Z_k(G)=G$ be the upper central series, we know that $G/Z_{k-1}(G)$ is nontrivial abelian. Moreover, it cannot be cyclic, since it is the central quotient of $G/Z_{k-2}(G)$ (and a group modulo its center cannot be nontrivial cyclic). Thus, $G/Z_{k-1}(G)$ certainly has such a surjection, which means $G$ itself does as well.