Let $G$ be an infinite cyclic group with generator $t$, and $M$ be a $G-$module. I need help with the following problem (of MacLane's book Homology) : Show that
$$H^{1}(G,M)\cong M/L$$
where $L=\{tx-x:x\in M\}$
My aproach: by definition is $H^{1}(G,M)=Der(G,M)/InnDer(G,M)$. It is natural to define $\rho:Der(G,M)\rightarrow M/L$ by:
$$\rho(\phi)=\phi(t)+L, \text{for all} \ \phi\in Der(G,M)$$
It is easy to proof that $\rho$ is an homomorphism of groups such that $Ker(\rho)=InnDer(G,M)$ but I can't see why $\rho$ is an epimorphism. If $v+L\in M/L$, for $v\in M$, it is natural to think that $v+L=\rho(\phi)$ where $\phi$ is determinated by $\phi(t)=tv=(t-1)v+v$ but I can't proof that $\phi$ is a derivation.
Note: the solution of the problem must to be obtain without use projective resolutions.