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.

Proof:

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.

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