Hm, I'm not familiar with the term "semidirect", what does it mean? I'm guessing you're demanding G to be abelian because of what semidirect means? Because if the theorem would say "direct sum" instead, then I don't see why we would need G to be abelian.
The semidirect product is a very handy generalization of the direct product. It is defined for every kind of group (not just abelians). If you're taking a course on group theory, then I'm pretty sure that this notion will pop up someday.
The semidirect product is, in general, a nonabelian group. The only situation when a semidirect product yields a abelian group, is when the semidirect product is in fact the direct product.
If you're interested, I suggest picking up a good group theory book and read about it. I recommend fully the book "The theory of finite groups" by Kurzweil and Stellmacher.
Hm, now I'm confused, why are direct products only defined for abelian groups?
If one defines [tex](a,\alpha) * (b,\beta) = (a*b,\alpha * \beta)[/tex] with a and b out of a certain group G and alpha and beta out of a certain group H, then we have a new group, don't we? (it's associative, has an inverse for every element and a neutral element)
Nonono, direct products are defined for any group, not just abelian groups. Semidirect products are also defined for all groups. Semidirect products are just a generalization of direct products.
What I said was, that if a semidirect product is abelian, then it had to be a direct product. That does certainly not mean that the direct product is always abelian or that it is only defined for abelian groups.
It's just that: the direct product of abelian groups is always abelian. But the semidirect product is never abelian (unless it was a direct product).
If you're confused, just forget everything I've said