Give an explicit definition for the different homomorphisms $\psi_1 ,\psi_2:\mathbb{Z}_{5}\to\mathbb{Z}_{5}$.

Question

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.

Answer 1

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 $\psi_i(1) = i$, $i \in \{ 1, 2, 3, 4\}$

Answer 2

The operation in ${\bf Z}_5$ is addition, so what you want is $$\psi(x+y)=\psi(x)+\psi(y)$$

You should

1. prove that $\psi(0)=0$,

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

3. see what happens when you take different values for $\psi(1)$.

Answer 3

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.