Suppose $f:G\longrightarrow H$ is a group homomorphism with $H$, $Ker(f)$ finite. Is $G$ finite? Let $f:G\longrightarrow H$ be a group homomorphism with $G$ not necessarily a finite group, but $H$ is a finite group. By the first isomorphism theorem we have:
$\frac{G}{Ker(f)}\cong Im(f)$.
Suppose further that we know that $Ker(f)$ is finite. Is it now possible to conclude that $G$ is a finite group? 
I am currently under the impression that lagrange's theorem can't be used, since it assumes the very thing we are trying to prove. Perhaps I am missing something obvious. Any help would be vastly appreciated. 

The kernel is one of the cosets in the quotient group and all cosets are the same size. Since the image is finite, there are a finite number of cosets. A finite number of cosets, each of a finite size implies that there are a finite number of elements in total.
