When is a group the direct sum of its normal subgroups?

  • #1
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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?
 

Answers and Replies

  • #2
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This doesn't really answer the question well, but if gcd(|N|,|G/N|)=1, then G is the semidirect product of N and G/N. This is the celebrated theorem of Schur-Zassenhaus.

So I guess, if gcd(|N|,|G/N|)=1 and if G is abelian, then [tex]G=N\oplus G/N[/tex].
 
  • #3
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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.
 
  • #4
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See http://en.wikipedia.org/wiki/Semidirect_product
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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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)
 
  • #6
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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 :smile:
 
  • #7
mathwonk
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  • #8
mathwonk
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it is useful to look at some simple examples. e.g. the integers Z have a subgroup 2Z which does not split this way. What do you think happens for the subgroup {0,3} of Z/6Z?
 

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