I was working on a homomorphism problem and had a quick question:
I am working with the group $G: \{-50, -49, \cdots, 47, 48, 49\}.$ The binary operator is, for $x, y \in G$, $x+y = (x+y) \pmod {100}.$
I thought of the following homorphism, $\phi: G \rightarrow G$:
$\phi(x) = \begin{cases} 0 &\mbox{if } x = 0 \\ 1 &\mbox{if } x \ne 0 \end{cases}$
I know one of the criteria for a homomorphism is that we must have $\phi(xy) = \phi(x)\phi(y)$. I was thinking that if we let $x = -50$ and $y = -1$, we get $\phi(50) = \phi(-50)\phi(-1)$. However, $50$ is not an element of $G$. Does this mean that the homomorphism is invalid or can I say that since $50 = -50 \pmod {100}$, $\phi(50)$ is just $\phi(-50)$?