Group operator vs. complex number multiplication operator in the multiplicative property of a character of a group

Question

A character $f$ of a group $G$ is defined as a complex-valued function defined on $G$ that 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.

I want to make sure I understand the definition of character right. I am going to make some claims per my understanding. Please tell me if my claims are true or false.

1. The group $G$ need not be a group of complex numbers. It may be a group of vectors, matrices or any other elements that satisfy the group postulates.
2. The domain of $f$ need not contain complex numbers.
3. The codomain of $f$ contains complex numbers.
4. In the notation $f(a)$, $a$ is an element that belongs to $G$ and the value of $f(a)$ is a complex number.
5. The notation $f(ab)$ denotes the function $f$ applied to $a \circ b$ where $ \circ $ represents the group operator, i.e. if $ \circ $ is $ + $, then $f(ab)$ denotes $f(a + b)$.
6. The notation $f(a)f(b)$ denotes complex number multiplication of $f(a)$ and $f(b)$.
7. Let $ a \in G $. Let $ \circ $ be the group operator. Let $ \times $ represent the multiplication operator to multiply complex numbers. The multiplicative property of the character can now be stated as $f(a \circ b) = f(a) \times f(b)$.

Are all these claims true or do you find some of them false?

Answer 1

Your statements are correct with one proviso. As the Wikipedia article Character theory states:

Let $V$ be a finite-dimensional vector space over a field $F$ and let $\rho:G\to \text{GL}(V)$ be a representation of a group $G$ on $V$. The character of $\rho$ is the function $\chi_\rho:G\to F$ given by $\chi_\rho(g) = \text{Tr}(\rho(g))$ where Tr is the trace.

It later states

A character of degree $1$ is called linear.

And also

However, the character is not a group homomorphism in general.

To summarize, your $\,f:G\to \mathbb{C^\times}\,$ is a group homomorphism which is of degree $1$ which is linear and corresponds to $\rho$. Since it is linear, then $\rho=\chi_\rho$ which is its own character. In general, a character can be equal to $0$, but linear characters can not take the value $0$. Your definition of character seems to be confusing the representation $\rho$ with its character $\chi_\rho$ which is easy to do if the degree is $1$.

For closely related but slightly different kind of characters compare group characters with the Wikipedia article on Dirichlet character defined on the multiplicative group of units of integers modulo $\,n.$