This theorem is pretty basic, but it is useful to construct non-abelian groups. Basically, once you have either group to be non-abelian, or the homomorphism to be trivial, the end result is non-abelian!
Theorem: The semidirect product is abelian iff , are both abelian and is trivial.
Assume is abelian. Then for any , , we have
This implies , thus is abelian.
Consider the case . Then for any , . Multiplying by on the left gives for any . Thus for all so is trivial.
Consider the case where . Then we have , so has to be abelian.
This direction is clear.