A remark from Wikipedia, http://en.wikipedia.org/wiki/Splitting_lemma
(about an exact sequence) $0\to A\to B\to C\to 1$
says " if a short exact sequence of groups is right split ... then it need not be left split or a direct sum ... what is true in this case is that B is a semidirect product, though not in general a direct product."
I cannot verify this, in particular showing that $AC = B$ is giving me problems as I don't know how to work with the right inverse (injection?) map $u:C\to B$.
If I define the map $\psi:AC\to B$ by $(a,c)\mapsto au(c)$, I want to show it is a bijection. But I can't even show it is well-defined. Is this the right bijection?