DIHEDRAL GROUP - Internal Direct Product

1. Sep 23, 2011

mehtamonica

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

Last edited: Sep 23, 2011
2. Sep 23, 2011

micromass

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

3. Sep 23, 2011

mehtamonica

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 ?

4. Sep 23, 2011

micromass

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

5. Sep 23, 2011

mehtamonica

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 ?

6. Sep 23, 2011

micromass

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

7. Sep 25, 2011

mehtamonica

Thanks a lot, Micromass.