# Thread: Isomorphism in External Product

1. ## Isomorphism in External Product

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

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

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

2. Originally Posted by tttcomrader
Is $S_{3} \oplus Z_{2} \approx Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, \_or\_ D_{6}$

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

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

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

3. Originally Posted by tttcomrader
Is $S_{3} \oplus Z_{2} \approx Z_{12}, Z_{6} \oplus Z_{2}, A_{4}, \_or\_ D_{6}$

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

Then, I can't really determine the rest, they are both non-cyclic, would that help?
You could look at orders of elements: for example, A_4 has eight elements of order 3 and three elements of order 2 (and the identity of order 1, of course).