DIHEDRAL GROUP - Internal Direct Product

[mehtamonica]

I have to prove that D4 cannot be the internal direct product of two of its proper subgroups.Please help.

 

[micromass]

So, what did you try already??
What can the orders be of a proper subgroup of D4?? Can they be abelian, nonabelian?

 

[mehtamonica]

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 ?

 

[micromass]

Indeed, an the direct product of abelian groups is...

 

[mehtamonica]

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 ?

 

[micromass]

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 $H\cap G=\{e\}$. Use that.

 

[mehtamonica]

Well, the internal direct product of H and G is isomorphic to the external direct product if $H\cap G=\{e\}$. Use that.

Thanks a lot, Micromass.