Prove if the the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not are isomorphic

Question

I'm trying to check if the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not isomorphic. How can I check if they are? I'm trying to understand how can I generally prove an isomorphism with this kind of groups. Any help?

Answer 1

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.

Answer 2

Here's one that kind of jumps out at me for this pair of groups. $S_3 \times \Bbb Z_2$ has a non-trivial center; in other words, it has a non-identity element that commutes with every element of the group. (Can you see what it is?) $A_4$ has no such element.