Everyone: I am trying to understand how to obtain a set of generators of a group $G$, given a homomorphism $h:G \to G'$ ($G'$ also a group); once we know the generators of $\ker(h)$ and $\mbox{im}(h)$ respectively. This is what I have so far: we get a SES:
$0 \stackrel{f_1}{\to} \ker(h) \stackrel{f_2}{\to} G \stackrel{f_3}{\to} G' \to 1$,
with
$f_1$ = only possible map.
$f_2$ = Identity map on $\ker(h)$
$f_3=h$, the given homomorphism
$f_4$ = The quotient map
But the sequence does not necessarily split that I know of. I imagine we need to use the fact that $G/\ker(h)$ is $h(G)$, the image of $h$, and maybe some property of Short-exact sequences that I don't know about. Any Ideas?
Thanks.
P.S: sorry for my lazyness in not yet having learnt Latex; thanks for the edit.