Why is $\phi(x^i)=y^i$ not a group homomorphism between the cyclic group of order $36$ to the cyclic group of order $17$? In Artin Algebra 2.10.3 there gives a group homomorphism from a cyclic group of order $12$ to a cyclic group of order $6$. Defined by $\phi(x^i)=y^i$ with $x$ in the cyclic group of order $12$ and $y$ in the cyclic group of order $6$. But we notice that for two cyclic groups with their order coprimes (for instance $36$ and $17$). This seems not to be a group homomorphism because we know that the only homomorphism between groups of coprime orders is the trivial homomorphism. 
Could someone tell me why it is not a group homomorphism with out us Lagrange's theorem, but just use the definition of homomorphism. Which condition is this map does not satisfy?
Thank for any help!

In general this will not even define a function between the groups, because the condition will require $\phi$ to take on different values at the same input.
(Notice, for instance, that if $G$ is cyclic of order 37 then there are many different ways of writing the identity: 
$$e = x^0 = x^{37} = x^{74} = x^{-37} = \cdots$$
Do these all give the same condition in general?)
