find which of the following groups is isomorphic to S3 Z2.
a) Z12 b) A4 c) D6 d) Z6 Z2
I eliminate option a because Z12 is cyclic whereas S3 Z2 is not because we know that the External direct product of G and H is cyclic if and only if the orders of G and H are relatively prime. Here it's not the case.
Here's my question. Can I eliminate option d using the following argument?
If S3 Z2 isomorphic to Z6 Z2 then we have S3 isomorphic to Z6, which is again a contradiction as Z6 is cyclic whereas S3 is not.
Is my argument right?
Also it would be great if I can get a head start with the other options too...