Prove that for every odd prime number $p$ there is a non-commutative group of order $p^3$ such that…

Question

Prove that for every odd prime number $p$ there is a non-commutative group of order $p^3$ such that $a^p = e$, $\forall a \in G$.

Answer 1

Here is the outline, you fill in the details:

Note that $\text{Aut}(\mathbb{Z}_p^2)\cong\text{GL}_2(\mathbb{F}_p)$, and since $|\text{GL}_2(\mathbb{F}_p)|=(p^2-1)(p^2-p)$ we see in particular, $p\mid |\text{Aut}(\mathbb{Z}_p)|$. Thus, there exists a non-trivial homomorphism $\varphi:\mathbb{Z}_p\to\text{Aut}(\mathbb{Z}_p^2)$. Consider then $G=\mathbb{Z}_p^2\rtimes_\varphi\mathbb{Z}_p$. Prove that this is the group you seek.