Give an explicit definition for the different homomorphisms $\psi_1 ,\psi_2:\mathbb{Z}_{5}\to\mathbb{Z}_{5}$. Can anyone please help me define clearly using the definition for a group homomorphism that for the given two homomorphisms it holds for all $g_1,g_2\in G$? 
We know that the definition gives that there exists two groups, $G$ and $H$ and a function  $\psi_1:G\to H$. This is then called a homomorphism if the equality $\psi_1(g_1+g_2)=\psi_1(g_1)+\psi_1(g_2)$ holds for all $g_1,g_2\in G$. 
So as a start,we take the first function $\psi_1:\mathbb{Z}_{5}\to\mathbb{Z}_{5}$. If we let $x,y\in\mathbb{Z}_{5}$, then $\psi_1(x+y)=\psi_1(x)+\psi_1(y)$?
Please brief me out on how to do this problem.
 A: The operation in ${\bf Z}_5$ is addition, so what you want is $$\psi(x+y)=\psi(x)+\psi(y)$$ 
You should 


*

*prove that $\psi(0)=0$, 

*prove that if you know $\psi(1)$ then you can work out $\psi(m)$ for all $m$, 

*see what happens when you take different values for $\psi(1)$. 
A: The group $\mathbb{Z}_5$ is written additively, so the homomorphism property spells as $\psi_1(x + y) = \psi_1(x) + \psi_1(y)$ in this case.
There are several ways to construct such a function. To find them all, try the following:
1) Which possible choices are there for $\psi_1(1)$?
2) Assure yourself, that by the homomorphism property all values $\psi_1(x)$ are determined once that a choice for $\psi_1(1)$ was made.
A: The group $\mathbb{Z}_5$ is an additive cyclic group: $(\mathbb{Z}_5, +)$, so the homomorphism property for $\psi_i: \mathbb{Z}_5 \to \mathbb{Z}_5$ is given by
$$\psi_i(x + y) = \psi_i(x) + \psi_i(y)$$
For all homomorphisms, $\psi_i(0) = 0$ (a homomorphism maps identity to identity). 
Then choose $\psi_i(1) = x \in \mathbb{Z}_5, x \neq 0$, the the homomorphism property guarantees that this assignment determines the assignment of all elements in $\mathbb{Z}_5$
For each element $x \in \mathbb{Z}_5$, $x \neq 0$, there exists a unique homomorphism determined by $\psi_i(1) = x$.
Hence, there are four such unique homomorphisms corresponding to $\psi_i(1) = i$, $i \in \{ 1, 2, 3, 4\}$
