Dihedral group - isomorphism

by mehtamonica
Tags: dihedral, isomorphism
 P: 26 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. 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 Dn= 2n = order of external direct product
 PF Patron Sci Advisor Thanks Emeritus P: 15,671 Try to show that Dn is nonabelian (for $n\geq 3$) and that your direct product will always be abelian...
P: 26
 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 !!

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