# Thread: Isomorphism in External Product

1. ## Isomorphism in External Product

Is $\displaystyle S_{3} \oplus Z_{2} \approx Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, \_or\_ D_{6}$

Now, S3 is not abelian, thus rule out both $\displaystyle Z_12$ and $\displaystyle Z_{6} \oplus Z_{2}$, which are abelian.

Then, I can't really determine the rest, they are both non-cyclic, would that help?

Is $\displaystyle S_{3} \oplus Z_{2} \approx Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, \_or\_ D_{6}$

Now, S3 is not abelian, thus rule out both $\displaystyle Z_12$ and $\displaystyle Z_{6} \oplus Z_{2}$, which are abelian.

Then, I can't really determine the rest, they are both non-cyclic, would that help?
Prove the following simple there.

$\displaystyle G$ and $\displaystyle H$ are abelian if and only if $\displaystyle G\times H$.

Is $\displaystyle S_{3} \oplus Z_{2} \approx Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, \_or\_ D_{6}$
Now, S3 is not abelian, thus rule out both $\displaystyle Z_12$ and $\displaystyle Z_{6} \oplus Z_{2}$, which are abelian.