Understanding of characters of a finite group.
Definition: Let, $G$ be any finite group. A complex valued function $f$ defined on $G$ is called a character of the group $G$ if $f$ has the multiplicative property $f(ab)=f(a)f(b)$ for all $a,b\in G$ and if $f(c)\ne 0$ for some $c\in G$.
We can show that for a finite group $G$ of order $n$ and for $a\in G$ , $f(a)$ is the $n$-th roots of unity. Also there are $n$-many characters.
Suppose, $|G|=5$. So there are $5$ characters of the group $G$. I want to find out all the $5$ characters explicitly. If $a\in G$ then $f(a)$ is the $5$-th roots of unity. i.e., $f(a)=e^{2k\pi i/5}$. How can I find the all characters ?
Edit :
All the characters of $\Bbb Z_5$ are $\displaystyle f_k(x)=e^{2k\pi ix/5}$ for $k=0,1,2,3,4$.
$f_0(x)=1$,for all $x\in \Bbb Z_5$, which is principal character.
Now, $\displaystyle f_1(0)=1, f_1(1)=e^{2\pi i/5}, f_1(2)=e^{4\pi i/5},f_1(3)=e^{6\pi i/5}, f_1(4)=e^{8\pi i/5}$.
Now, $f_1(2)f_1(3)=e^{2\pi i}$, but $f_1(2.3)=f_1(1)=e^{2\pi i/5}$.
So $f_1(2.3)\ne f_1(2)f_1(3)$. That contradicts our definition. That is the fact which is not clear to me. Can you please explain this where my mistake?
In general, if $G$ is a cyclic group of order $n$, with generator $a$, then the characters have the form $\chi(a^k)=\zeta^k$ where $\zeta$ is an arbitrary $n$-th root of unity.