Say a sequence $A\overset{f}{\to}B\overset{g}{\to}C$ of group morphisms is coexact if $\operatorname{Coker}f= \operatorname{Im}g$.
The first isomorphism theorem ensures that exactness implies coexactness. In an abelian category they coincide (because the property of being abelian is autodual). Where do they differ?
Question. What are some examples of coexact sequences of groups morphisms which are not exact?
I think we want the normal closure of $\operatorname{Im}(f)\leq B$ to be strictly smaller than $\operatorname{Ker}g$, but I'm really bad with all this normal stuff.
Let $A$ be a non-normal subgroup of $B$ and let $N$ be the normal subgroup it generates. Then the sequence $A\to B\to B/N$ is coexact: the cokernel of $A\to B$ is the universal morphism out of $B$ that annihilates $A$, which is just the quotient $B\to B/N$.
More generally, coexactness of $A\overset{f}{\to}B\overset{g}{\to}C$ means precisely that the image of $f$ generates the kernel of $g$ as a normal subgroup, since the cokernel of $f$ is exactly the quotient by normal subgroup generated by its image.