Let $F: \mathbb{Z}_{20} \to \mathbb{Z}_{20}$ defined by $F(x) = 14x$. Prove that $F$ is an homomorphism, determine the kernel of F and the image $F(\mathbb{Z}_{20})$.
I barely understand the concepts and this is the first problem I handle of this magnitude. I just want a step by step of what is happening. I'm having trouble in this subject and I suppose this is an easy level exercise for me to learn about and then analyze and digest it. Any help would be great.
Best Answer
To prove it is a group homomorphism in $(\mathbb{Z_{20}}, +)$, you need to show that $F$ preserves the sum operation, that is $F(x+y)=F(x)+F(y)$, which is true because the ring $\mathbb{Z_{20}}$ has the distributive property.
The kernel of $F$ is $Ker(F):=\{x\in \mathbb{Z_{20}}\mid F(x)\equiv 0\}$, that is every integer $x$ such that $20$ divides $14x$, which are all multiples of $10$ (the ideal $\langle \overline{10}\rangle)$, because $20\mid 14x\Leftrightarrow 10\mid 7x\Leftrightarrow 10\mid x$, given that the $gcd(10,7)=1$.
For the image, use some of the classical isomorphism theorems and you'll get $Im(F)\simeq\mathbb{Z_{20}}/Ker(F)=\mathbb{Z_{20}}/\langle \overline{10}\rangle \simeq \frac{\mathbb{Z}/\langle 20\rangle}{\langle 10\rangle/\langle 20\rangle} \simeq \mathbb{Z_{10}}$.
You should check the so-called first and third isomorphism theorems for groups to see why they are usable here.