Which of these groups are isomorphic?

Question

A computer program outputs the following Cayley tables for groups of order 4.

Wikipedia tells me that there are only two groups of order 4, the cylic group ($Z_4$) and the Klein four-group ($Dih_2 = Z_2 \times Z_2$), so some of these groups must be isomorphic.

 * [e][a][b][c]
[e] e  a  b  c  
[a] a  e  c  b  
[b] b  c  e  a  
[c] c  b  a  e  

 * [e][a][b][c]
[e] e  a  b  c  
[a] a  e  c  b  
[b] b  c  a  e  
[c] c  b  e  a  

 * [e][a][b][c]
[e] e  a  b  c  
[a] a  c  e  b  
[b] b  e  c  a  
[c] c  b  a  e  

 * [e][a][b][c]
[e] e  a  b  c  
[a] a  b  c  e  
[b] b  c  e  a  
[c] c  e  a  b  

Which of these groups are isomorphic and why?

Answer 1

An isomorphism can be interpreted as a relabelling of the elements (except for the identity $e$ which is fixed by an isomorphism). So, if you can find two tables which are the same after interchanging the roles of $a, b, c$ then you will have the multiplication tables of isomorphic groups. For example, the third table is the same as the second if you replace $c$ by $a$ and $a$ by $c$. Therefore, these two tables correspond to isomorphic groups.

Suppose now that you're willing to take as given that there are only two groups of order four: $\mathbb{Z}_4$ and $\mathbb{Z}_2\times\mathbb{Z}_2$. Note that $\mathbb{Z}_4$ has one element of order two, but $\mathbb{Z}_2\times\mathbb{Z}_2$ has three. As $e^2 = e$, we can tell which group corresponds to each table by counting how many times $e$ appears on the diagonal, two for $\mathbb{Z}_4$ and four for $\mathbb{Z}_2\times\mathbb{Z}_2$.

Answer 2

You can see that in the first one, every non-identity element $x$ is of order 2, i.e., $x^2 = e$.

This is not true in the other three, each of which contains one element of order 2. Those three have to be isomorphic to $\mathbb{Z}_4$. Why? Well, because it is a fact that there are only two groups of order 4, and the Klein four-group is exactly that group of order four with all non-identity elements having order 2. So the last three groups have to be isomorphic to $\mathbb{Z}_4$ (if they are groups). Of course, you can meticulously verify this.

Answer 3

HINT: Look at the diagonal. A diagonal entry is $e$ if and only if the corresponding element has order two.