I am studying representation theory, and below is an excerpt from my lecture notes.
Suppose that $G = (\Bbb Z,+)$. Then, if $\rho$ is a representation of $G$, it is completely determined by $V$ and the invertible linear map $\rho(1) : V\to V$ (which can be anything). This is because we then have $$\rho(n) = \rho(1 + \dots + 1) = \rho(1)^n.$$ Thus a representation of $\Bbb Z$ is just a vector space $V$ together with an invertible linear map from $V$ to itself.
I am confused as to the middle line, where $\rho(n) = \rho(1 + \dots + 1) = \rho(1)^n$. I would have thought that $\rho(n) = \rho(1 + \dots + 1) = \rho(1) + \dots + \rho(1) = n\rho(1).$ Why is this as so? Thank you in advance :)
The elements of the codomain are just written multiplicatively (because they're functions).