Let $p$ be a prime. Let $H_{i}, i=1,...,n$ be normal subgroups of a finite group $G$. I want to prove the following: If $G/H_{i}$, $i=1,...,n$ are abelian groups of exponent dividing $p−1$, then $G/N$ is abelian group of exponent dividing $p−1$ where $N=\bigcap H_{i},i=1,...,n$.
Proof: Since $G/H_{i}$, $i=1,...,n$ are abelian groups, then $G^{′}$ (the derived subgroup of $G$) is contained in every $H_{i}, i=1,...,n$. Hence $G^{′}$ is contained in $N$. Therefore $G/N$ is abelian. I do not know how to deal with the exponent.
Thanks in advance.