I am trying to prove that given a homomorphism from group $G$ to group $G'$ $f: G \to G'$,
- If $G$ is finite then $|im(f)|$ is finite and divides $|G|$
- If $G'$ is finite then $|im(f)|$ is finite and divides $|G'|$
I know $|im(f)|$ is finite as it can be at most $|G|$, also finite. I know $ker(f) \leq G$ so $|ker(f)|$ divides $|G|$ by Lagrange's theorem.
I also know that the cosets of the kernel are the subsets of $G$ whose elements are mapped to the same element in $im(f)$. I need help for the first one as I cant seem to make a well-defined bijective function with the collection of cosets of $ker(f)$ as a domain and the image as a codomain. Do I need a function about the image and the kernel and make the function utilize the cosets and use counting somehow instead? I'm new to cosets so I have trouble utilizing them. I know that the order is the same for the kernel and its cosets but is using a function from an arbitrary coset of the kernel to the image of f somehow easier? I am mind-boggled at this point.
For the second one I know that $im(f)$ is a subgroup of $G'$, so $im(f) $ is a subset of $G'$ and is also finite so by lagranges thm $|im(f)|$ divides $|G'|$ But if I am wrong please point it out.