Hello all I have a question on group theory.
Let $G$ be any group and $H$ a finite normal subgroup of $G$. Suppose that the quotient $G/H$ is abelian. Is it true then that $G$ is abelian? If not, do you have a counterexample?
My attempt: I think the answer is positive, as taking only a finite piece of $G$ does not affect very much on its behavior.
Thanks in advance!
*Edit: I edit my question as I received counterexamples with finite groups (which I really appreciate). What if we add the assumption that $G$ is infinite?