# Math Help - Isomorphism of Direct product of groups

1. ## Isomorphism of Direct product of groups

find which of the following groups is isomorphic to S3 $\bigoplus$ Z2.

a) Z12 b) A4 c) D6 d) Z6 $\bigoplus$ Z2

I eliminate option a because Z12 is cyclic whereas S3 $\bigoplus$ 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 $\bigoplus$ Z2 isomorphic to Z6 $\bigoplus$ 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...

Thanks

2. ## 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.

3. ## 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.

4. ## 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.

$S^3$ is a subgroup of $S^3\times Z_2$. Can you realise $S^3$ as a subgroup of either of your two remaining options?

5. ## Re: Isomorphism of Direct product of groups

$S_{3}\oplus Z_{2}$ is not isomorphic to $A_{4}$ because the element ((123),1) has order 6 while $A_{4}$ doesn't have any element of order 6. Actually $S_{3}\oplus Z_{2}$ is isomorphic to the dihedral group $D_{12}$.

6. ## Re: Isomorphism of Direct product of groups

xixi, your justification was very elegant. it took a while to strike me as to why A4 should n't have an element of order 6, i realized that the order of any element of A4 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 D6, eh?

7. ## Re: Isomorphism of Direct product of groups

Yes, $S_{3} \oplus Z_{2}$ is isomorphic to $D_{12}$ (It is the dihedral group of order twelve) which though denoted $D_{6}$ in an alternate convention.In other words, it is the dihedral group of degree six.