Sufficient Condition for Quotient Group by Intersection to be Abelian

Theorem
Let $G$ be a group.

Let $N$ and $K$ be normal subgroups of $G$.

Let the quotient groups $G / N$ and $G / K$ be abelian.

Then the quotient group $G / \paren {N \cap K}$ is also abelian.

Proof
From Intersection of Normal Subgroups is Normal, we have that $N \cap K$ is normal in $G$.

We are given that $G / N$ and $G / K$ are abelian.

Hence: