Determine if $\Bbb Z_2 \times \Bbb Z_3$ is isomorphic to $\Bbb Z_6$.

Question

Determine if $\Bbb Z_2 \times \Bbb Z_3$ is isomorphic to $\Bbb Z_6$.

Computing the direct product I have that $\Bbb Z_2 \times \Bbb Z_3 = \{(x,y) \mid x \in \Bbb Z_2, y \in \Bbb Z_3 \} =\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)\}$ this has $6$ elements as does $\Bbb Z_6 = \{0,1,2,3,4,5\}$ I used this to conclude that the two groups are isomorphic, but I have a feeling that this doesn't count as a way to determine this. It seems that this could be done by looking at the orders of each group? I know that the order of a $g \in G$ is the smallest $n$ for which $g^n = e$, but I don't see how this can be used to show isomorphism?

Answer 1

Yes, they are isomorphic, but having the same number of elements is not enough for that. Afer all $S_3$ also has $6$ elements, but it's not isomorphic to any of those two groups, which are abelian.

Simply define$$\begin{array}{rccc}\varphi\colon&\Bbb Z_6&\longrightarrow&\Bbb Z_3\times\Bbb Z_2\\&[n]&\mapsto&(n\pmod 3,n\pmod 2)\end{array}$$(that is, $\varphi(0)=(0,0)$, $\varphi(1)=(1,1)$, $\varphi(2)=(2,0)$, …). Now, prove that $\varphi$ is an isomorphism.

Answer 2

By the Chinese remeinder Theorem (https://en.wikipedia.org/wiki/Chinese_remainder_theorem), you have $\mathbb Z_p \times \mathbb Z_q \cong \mathbb Z_{pq}$ if and ony if $\gcd(p,q)=1$.

Answer 3

There is an isomorphism $\phi:\mathbb{Z}_2\times \mathbb{Z}_3\to \mathbb{Z}_6$.
Send $([1]_2,[1]_3)\mapsto [1]_6$ and $([0]_2,[0]_3)\mapsto [0]_6$, you can find out the mapping in detail from here, and then verify this is an isomorphism.