Hi,

Someone told me that a good exercise to do to get better understanding of groups is proving that S3 and D3 (the dihedral group of order 6) are isomorphic. Is there any other way to do that than explicitly constructing a function from one group to the other and show that it is a bijective homomorphism or writing out the multiplication tables and seeing they match?

I remarked that each group as one element of order 1 (the identity), 2 elements of order 3 and 3 elements of order 2. Is this sufficient to show that the groups are isomorphic i.e would it be possible to find a group of order 6 with exactly the same number of elements with these orders that is not isomorphic to S3 or D3?