Okay, so I'm working on a homework problem in abstract algebra, and I have found the solution already, what I want to know is why my initial line of reasoning didn't work - i..e, what have I done or assumed that I am not allowed to do.
The problem is as follows. Suppose that $\phi: Z_{50} \rightarrow Z_{15}$ is a homomorphism of groups, such that $\phi (7)=6$ Determine $\phi (x)$ for every $x \in Z_{50}$. Here was my line of reasoning.
$ \phi (7)=6 \implies [\phi (7)]^2=6^2 \implies \phi(49)=6$ by the properties of a homomorphism (which I'm reading from Galian, 8th ed. page 210, in case that matters). But $49 \equiv -1 \pmod{50}$, so that $\phi(-1)=6$. But now we can use the same property as before to note that $[\phi(-1)]^2 = \phi(1)=6.$ which would imply that $\phi (x) = 6x$.
But something in this line of reasoning is wrong, because $7\times 6$ is 42, which is not $6 \bmod 15$. It's $12$.
Now, I already know that the right definition for $\phi$ is that $\phi (x)= 3x$, as $3$ is the smallest integer $a$ so that $7a\equiv 6 \pmod{15}$, what I don't know and can't figure out, is why my analysis is wrong the first time.
Because the operation in $\mathbb{Z}_{50}$ is supposed to be the sum and not the product… it is not a ring homomorphism, but just a group homomorphism and those are groups with respect to the sum! A group homomorphism preserves only an operation and in this case it is the sum, not the product!