Isomorphism between Dihedral and Symmetric groups of the same order?

[Bleys]

Is there a way to prove generally that the Dihedral group and its corresponding Symmetric group of the same order are isormorphic. In class we were only shown a particular example, D3 (or D6 whatever you wish to use) and S3, and a contructed homomorphism, but how could you do it generally? Would you still have to construct a specific map and show that it's a bijective homomorphism? Or can you just simply show there exists at least one isomorphic map between the two?

 

[CompuChip]

They are not. In general, D_n (or D_2n) has 2n elements, whereas S_n has n!.

Of course, the reason that it works for the triangle group, is that any permutation of its vertices is also a symmetry. However, the permutation group for the vertices of a square is already larger than its symmetry group. For example: if you label the corners 1, 2, 3, 4 in clockwise order, then the symmetry that interchanges 1 with 3 and 2 with 4 does not correspond to any element from D₄ (or D₈).