Isomorphism of Direct product of groups

find which of the following groups is isomorphic to S_{3} Z_{2}.

a) Z_{12} b) A_{4} c) D_{6} d) Z_{6} Z_{2}

I eliminate option a because Z12 is cyclic whereas S_{3} Z_{2} 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 S_{3} Z_{2 }isomorphic to Z_{6} Z_{2} then we have S_{3} isomorphic to Z_{6}, which is again a contradiction as Z_{6} is cyclic whereas S_{3} is not.

Is my argument right?

Also it would be great if I can get a head start with the other options too...

Thanks

Re: Isomorphism of Direct product of groups

An easier argument for d) is that it is abelian, whereas your original group is not. I don't think your original argument is justified.

Re: Isomorphism of Direct product of groups

Hi Gusbob,

I have found justification for my claim, yet I agree with you that your argument that the property of "being abelian" is a more convincing and more elegant solution. Any ideas about the other two options ... I just need to eliminate one more to arrive at the answer.

Re: Isomorphism of Direct product of groups

The most obvious hint is a giveaway, but I can't think of anything else at an elementary level short of writing an explicit isomorphism to the correct answer.

is a subgroup of . Can you realise as a subgroup of either of your two remaining options?

Re: Isomorphism of Direct product of groups

is not isomorphic to because the element ((123),1) has order 6 while doesn't have any element of order 6. Actually is isomorphic to the dihedral group .

Re: Isomorphism of Direct product of groups

xixi, your justification was very elegant. it took a while to strike me as to why A_{4} should n't have an element of order 6, i realized that the order of any element of A_{4 }is got to be the lcm of the cycles into which it can be split, which can never exceed 4 cos splitting 4 letters can only be done with at most 4 parts or lesser.

still i guess you mean to say that the answer is D_{6}, eh?

Re: Isomorphism of Direct product of groups

Yes, is isomorphic to (It is the dihedral group of order twelve) which though denoted in an alternate convention.In other words, it is the dihedral group of degree six.