DIHEDRAL GROUP - Internal Direct Product

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Discussion Overview

The discussion revolves around the properties of the dihedral group D4 and the challenge of proving that it cannot be expressed as the internal direct product of two of its proper subgroups. The scope includes theoretical exploration of group properties, subgroup orders, and the nature of direct products.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks assistance in proving that D4 cannot be the internal direct product of two of its proper subgroups.
  • Another participant questions what has been attempted so far and discusses the possible orders of proper subgroups of D4, suggesting they could be abelian or nonabelian.
  • A participant notes that if G is the internal direct product of subgroups H and K, the possible orders of H and K could be 2 and 4, or vice versa, and expresses uncertainty about how to proceed given that both H and K seem to be abelian.
  • There is a discussion about whether the internal direct product of abelian subgroups H and K can be concluded to be abelian as well.
  • Another participant mentions that the internal direct product of H and G is isomorphic to the external direct product if the intersection of H and G is trivial.

Areas of Agreement / Disagreement

Participants express uncertainty about the properties of the internal direct product in relation to the abelian nature of the subgroups. There is no consensus on whether D4 can be expressed as an internal direct product of its proper subgroups, and multiple viewpoints are presented regarding subgroup properties.

Contextual Notes

Limitations include the lack of specific examples or counterexamples regarding the subgroup structure of D4, as well as unresolved questions about the implications of subgroup orders and their abelian properties.

mehtamonica
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I have to prove that D4 cannot be the internal direct product of two of its proper subgroups.Please help.
 
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So, what did you try already??
What can the orders be of a proper subgroup of D4?? Can they be abelian, nonabelian?
 
micromass said:
So, what did you try already??
What can the orders be of a proper subgroup of D4?? Can they be abelian, nonabelian?
Thanks, Micromass. If G is the internal direct product of its subgroups H and K ,then the possible orders of subgroups H and K can be 2 and 4 or vice a versa.

It seems that both H and K are abelian. How can move further from this ?
 
Indeed, an the direct product of abelian groups is...
 
micromass said:
Indeed, an the direct product of abelian groups is...

As far as the result goes the external direct product of two abelian groups is abelian...but is the internal direct product abelian too ? i mean if subgroups H and K are abelian can we conclude that the IDP is abelian ?
 
mehtamonica said:
As far as the result goes the external direct product of two abelian groups is abelian...but is the internal direct product abelian too ? i mean if subgroups H and K are abelian can we conclude that the IDP is abelian ?

Well, the internal direct product of H and G is isomorphic to the external direct product if [itex]H\cap G=\{e\}[/itex]. Use that.
 
micromass said:
Well, the internal direct product of H and G is isomorphic to the external direct product if [itex]H\cap G=\{e\}[/itex]. Use that.

Thanks a lot, Micromass.
 

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