Let $M$, $N$ be normal subgroups of a group $G$.
Prove that if $G/M\cap N$ is cyclic, then $G/M$ and $G/N$ are cyclic. Give a counter example to show that the converse is not always true.
I proved until now that if $G/M\cap N$ is abelian, then $G/M$ and $G/N$ are abelian, I don't know if that helps a lot but it's the only thing I could come up with.
Best Answer
Hint for $\implies$:
Any quotient of a cyclic group is a cyclic group.
The third isomorphism theorem.
Hint for $\;\;\,\not\!\!\!\!\impliedby$: