I am considering the one-to-one homomorphism $\phi:G \rightarrow H$.
If I take some element $g$ of $G$ and take its order as $n$ then $n$ is the smallest positive integer such that $x^n=e$. Now I consider the homomorphic image $\phi(g)$ of $g$ and take its order to be $m$ then $(\phi(g))^m=e'$ where $e'$ is the identity element of $H$.
Since $\phi$ is a homomorphism, it could be written as $(\phi(g^m))=e'=\phi(e)$ as $\phi$ is $1-1$ as well. Thus gives us $g^m=e$. Now this means that order of $g$ i.e. $n$ divides $m$ which is order of its homomorphic image. But in books, it is given the other way round i.e. order of homomorphic image of an element divides the order of element. So where am I going wrong here? Please suggest.