# Dihedral group - isomorphism

1. Aug 26, 2011

### mehtamonica

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

2. Aug 26, 2011

### micromass

Try to show that Dn is nonabelian (for $n\geq 3$) and that your direct product will always be abelian...

3. Aug 31, 2011

### mehtamonica

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 !!