group $G$ such that $|G| = p^k$ for some prime $p$ and some integer $k > 1$, where $|g| = p$ for every $g \in G \setminus {\epsilon}$

Question

Question:
Give an example of a group $G$ such that $|G| = p^k$ for some prime $p$ and some integer $k > 1$, where $|g| = p$ for every $g \in G \setminus {\epsilon}$.
My Reasoning :
I wasn't too sure how to approach this question. Would the group $\mathbb{Z}_2 \times \mathbb{Z}_2$ be such a group? I wasn't too sure because the operation of this group is addition and not multiplication so I can't take powers of elements, which is what this question asks. for example : $(1,0)^2 = (1,0)$ $\neq$ $(0,0)$. I also thought that I can just take the general Klein group where: $V = \{1, a,b,c\mid a^2=b^2=c^2\}$. Are these two groups good examples? if not, what other group can I pick?

Answer 1

More generally: $$G=C_p\times C_p$$ where $C_p$ is the cyclic group of order $p$ (in multiplicative notation), has those properties: it has order $p^2$ and every nontrivial element has order $p$. Your example (one, actually, as $\Bbb Z_2\times \Bbb Z_2\cong V$) is just the special case for $p=2$. So, yes, it is a good example.