Why can't Dn be isomorphic to the direct product of its subgroups?

[mehtamonica]

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

Please suggest how to go about it.

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 D_n= 2n = order of external direct product

 

[micromass]

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

 

[mehtamonica]

Try to show that D_n 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 !