$p$-group problem

Question

Let $A,B,C$ are three subgroups in a way that $1<A \triangleleft B \triangleleft C$. With $B/A$ and $C/B$ are $p$-groups. Then prove that $|C|$ is also a $p$-group. I have been trying to prove it for hours but can't find a way to solve it. So any help will be appreciated.

Answer 1

The result is false. Let $A=\mathbb{Z}_2 \times \{0\} \times \{0\}$, $B=\mathbb{Z}_2 \times \mathbb{Z}_3 \times \{0\}$, and $C=\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_3$. $B/A$ and $C/B$ are each isomorphic so $\mathbb{Z}_3$ (so $3$-groups), but $|C|$ is not a power of a prime so $C$ is not a $p$-group for any $p$, let alone $p=3$.