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## Main Question or Discussion Point

I was sad to find out that if H is a normal subgroup of G, we can't say [tex]G \cong H \oplus G/H[/tex]. Now I'm wondering: in which cases does this equality hold?

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I was sad to find out that if H is a normal subgroup of G, we can't say [tex]G \cong H \oplus G/H[/tex]. Now I'm wondering: in which cases does this equality hold?

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So I guess, if gcd(|N|,|G/N|)=1 and if G is abelian, then [tex]G=N\oplus G/N[/tex].

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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.

- #5

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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)

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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

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mathwonk

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the free notes 3.a, page 47, and 6.d p. 6, discuss semi direct products.

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mathwonk

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