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!

Taking directly your example
$$\phi:C_{36}=\langle x\rangle\to C_{17}=\langle y\rangle\;,\;\;\phi(x^k)=y^k$$
For example we'd get
$$1=\phi(x^{36})=\phi(x^{17}x^{19})=\phi(x^{17})\phi(x^{19})=y^{17}y^{19}=1\cdot y^{19}=y^2\implies y^2=1$$
which of course is absurd.
