Let’s say I have two groups $G=\mathbb Z_{12}$ and $H=\mathbb Z_2^2×\mathbb Z_3$. I want to check if the two groups are isomorphic or not.
I clearly know that $G$ is cyclic. I consulted the answer sheet and the solution is something like this:
The two groups are not isomorphic because while all the elements of $H$ have order at most 6, $G$ has an element of order 12.
My question is how do I know the that the former is true? I guess checking with every element is one of the ways, but it can be really tedious.
$Z_{mn} \cong Z_{m} \times Z_{n}$ iff $(m,n)=1$, that means greatest common divisor of $m$ and $n$ is one. Please note one more thing that isomorphism does not preserve order only it is much more than just preserving order.