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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

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up vote 7 down vote accepted

${D}_n \not\cong \mathbb{Z}/2 \oplus \mathbb{Z}/n,$ as the latter is Abelian and the former is not.

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