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Isomorphism of Dihedral D2m to Dm x Z2

For odd m>=3, prove that D2m is isomorphic to Dm x Z2. Where D is the Dihedral group and Z is the group of integers.

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For any m, D_m can be represented by rotations in complex plane:

D_m(k) = { e^{2 pi i k / m}, k = 0, 1, ...., m-1 }

and Z_2 can be represented by D_2:

Z_2 = { e^0, e^{pi i} }

If m is odd, the direct product D_m x Z_2 can be represented 1-to-1 by rotations which are the usual products of ...