Prove two groups are isomorphic

Question

I have created the two tables but can not find a one to one correlation between the values in the two tables. I would appreciate it if anyone can point me in the right direction to understand how to solve this.

+4  0   1   2   3  
0   0   1   2   3  
1   1   2   3   4  
2   2   3   0   1  
3   3   4   1   2  

*5  1   2   3   4  
1   1   2   3   4  
2   2   4   1   3  
3   3   1   4   2  
4   4   3   2   1

Thank you!

question 31.3-1

Answer 1

The groups are cyclic; that means they are generated by a single element.

A generator of $\Bbb Z_4$ is $1$; i.e., the elements in $\Bbb Z_4$ are $0, 1, 1+1, $ and $1+1+1$.

A generator of $\Bbb Z_5^*$ is $2$; i.e., the elements in $\Bbb Z_5^*$ are $1, 2, 2^2, 2^3\equiv_53. $

The map from $\Bbb Z_4$ to $\Bbb Z_5^*$ that takes $0$ to $1$, $1$ to $2$, $1+1$ to $2\times2$, and $1+1+1$ to $2^3\equiv_5 3$

is an isomorphism.

To prove that from scratch, you would have to verify $16$ statements

(actually only $10$ if you use commutativity).