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Dihedral group  isomorphism 
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#1
Aug2611, 12:34 PM

P: 26

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 


#2
Aug2611, 01:44 PM

Mentor
P: 18,036

Try to show that D_{n} is nonabelian (for [itex]n\geq 3[/itex]) and that your direct product will always be abelian...



#3
Aug3111, 04:24 PM

P: 26

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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