Here's a question that I came across in my textbook. It says:
Decide whether the following group is isomorphic to $(\mathbb{Z_4},+)$ or K, where K is the Klein $4$-group.
I am given the following set:
$$\{Id,(12),(34),(12)(34)\}$$
I know that, as a function, this comes out to be $f(x) = x_1+x_2-x_3-x_4$. I also know that if that, in general, a group G is a group of order 4, then it is isomorphic to either $(\mathbb{Z_4},+)$ or K. So how do I determine whether my given set is isomorphic to $(\mathbb{Z_4},+)$ or K? I believe that I am suppose to find an element such that when I raise it to the $4^{th}$ power, I get my identity matrix, which is Id. That's as far as I can go with this one.
Once we have established that our set $S=\{\operatorname{id}, (12),(34),(12)(34)\}$ is indeed a group, why don't we just prove your general statement: a group of order 4 is either isomorphic to $(\mathbb{Z}_4,+)$ or $K$?
For the first, we need to check if $S$ is closed under composition and the inclusion of inverses. We consider the action of its elements on the set $\{1,2,3,4\}$ to simplify the compositions.
For the second, we compare their cyclic- and multiplication tables, in order to construct a group isomorphism $a:S \to(\mathbb{Z}_4,+)$ or $b: S\to K$.