Quotient Group by Intersection of Normal Subgroups not necessarily Cyclic if Quotient Groups are

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Let $G$ be a group.

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

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

Then the quotient group $G / \paren {N \cap K}$ is not necessarily cyclic.


Proof by Counterexample:

Let $G = \set {e, a, b, c}$ be the Klein $4$-group whose identity element is $e$.

Let $N = \set {e, a}$ and $K = \set {e, b}$.

By Subgroups of Klein Four-Group, both $N$ and $K$ are subgroups of $G$.

By Prime Group is Cyclic, both $N$ and $K$ are cyclic.

By Subgroup of Abelian Group is Normal, both $N$ and $K$ are normal in $G$.

Then we have that:

$N \cap K = e$

and so by Trivial Quotient Group is Quotient Group:

$G / \paren {N \cap K} \cong G$

But $G$ is the Klein $4$-group, which is not cyclic.