## Dihedral group - isomorphism

The dihedral group Dn of order 2n has a subgroup of rotations of order n and a subgroup of order 2. Explain why Dn cannot be isomorphic to the external direct product of two such groups.

If H denotes the subgroup of rotations and G denotes the subgroup of order 2.

G = { identity, any reflection} ( because order of any reflection is 2)

I can see that order of Dn= 2n = order of external direct product
 PhysOrg.com science news on PhysOrg.com >> Intel's Haswell to extend battery life, set for Taipei launch>> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus Try to show that Dn is nonabelian (for $n\geq 3$) and that your direct product will always be abelian...

 Quote by micromass Try to show that Dn is nonabelian (for $n\geq 3$) and that your direct product will always be abelian...
Thanks a lot. I got it.

If we take H as the subgroup consisting of all rotations of Dn, then being a cyclic group, it would also be abelian. Then again, subgroup K of order 2 is abelian.

Further, the external direct product H + K is abelian as H and K are abelian.

Thanks !!