Quotient groups of $p$-groups

Question

Suppose I am trying to show that a group $G$ is solvable and I gotten to having $Z(G)$ be a p-group and $G/Z(G)$. Now if I can show that $G/Z(G)$ is also a $p$-group, then both are solvable implying $G$ is solvable. Now intuitively it makes sense that $G/Z(G)$ is a p group, but how can one show it? Thanks in advance.

Answer 1

$G$ is a $p-$group, implies that $|G|=p^n$ for some positive integer $n$. $Z(G)$ is non-trivial and $Z(G)$ is also a $p-$group, implies that $|Z(G)|=p^k$, where $1\leq k \leq n$. If $n=k$, then $G$ is an abelian group, so solvable. If $n\neq k$, then $|G/Z(G)|=|G|/|Z(G)|=p^{n-k}$. Therefore $G/Z(G)$ is also a $p-$group.