Indeed, they do have a one-to-one correspondence between, them (in fact, there are $8!$ such correspondences). The problem is that they don't have the "same structure," so none of these correspondences will be homomorphisms, and so no isomorphism exists. But it's tedious in the extreme to check so many functions to see if any of them is a homomorphism, so we instead make use of one of many properties preserved under isomorphisms. Two such (related) properties are that:
- isomorphisms preserve orders of elements (meaning that any given element will be mapped to an element of the same order), and
- isomorphic images of cyclic groups are again cyclic groups.
Those two properties are probably the simplest way to show that these two groups are not isomorphic, since (in particular) there are four different order $8$ elements of $\Bbb Z_8,$ but no such elements in $\Bbb Z_2\times\Bbb Z_4.$ So, we can directly conclude that they are not isomorphic by fact 1. Alternately, we can note that this means $\Bbb Z_8$ is cyclic and $\Bbb Z_2\times\Bbb Z_4$ is not, so can indirectly conclude that they are not isomorphic by fact 2.
There are many other such properties that we can use in other cases. It's good to keep several of them in mind. As Mark points out in the comments, though, the first property is a necessary condition for a function to be an isomorphism, but not sufficient. However, two cyclic groups of the same order will necessarily be isomorphic. Unfortunately, most groups aren't cyclic, so this property won't usually be useful.
$$G=S_3\times\mathbb{Z_2}$$ is not abelian and not cyclic hence it's not isomorphic to $\mathbb{Z_{12}}$ and $\mathbb{Z_6}\times\mathbb{Z_2}$. Also the group $A_4$ doesn't have an element of order $6$ but $G$ has, say $\{(1,2,3),1\}$, so correct option is $D_6$
Best Answer
When proving that two groups aren't isomorphic, you need to find some property which shows they are different. The easiest one is the number of elements. Unfortunately they both have $12$ elements, so we're out of luck there.
The next step in the same direction is to look at the orders of elements: How many elements do each of the two groups have of order $2$? How many elements of order $3$? $4$? $6$? $12$? Is there any of those for which our groups are different? If yes, then the groups cannot be isomorphic.
As you keep learning about group theory, you'll learn about more things you can use to to differentiate groups: The structure of subgroups, of normal subgroups, the center, the derived subgroup, and so on.