According to Silverman(Arithmetic of Elliptic curves):
The definition of a crossed homomorphisms, is a map $f : G \to M$ satisfying $f(ab)=bf(a)+f(b)$ for all $a$, $b$ in G.
According to many other books:
The definition of a crossed homomorphisms, is a map $f : G \to M$ satisfying $f(ab)=f(a)+af(b)$ for all $a$, $b$ in G.
Whether both the definitions are the same or different? If both the definitions are same then how I can define an action of $G/N$ on the first cohomology group $H^1(N,A)$.(where $N\trianglelefteq G$)
Proof. From the other answer you have
$$(g\cdot f)(h)=gf\left(g^{-1}(hg)\right) =gf(g^{-1})+f(hg)=gf(g^{-1})+f(h)+hf(g).$$
Using this equality we get
$$ \begin{align} \ h_{1}\left( g \cdot f\right) \left( h_{2}\right) &=h_{1}[gf\left( g^{-1}\right) +f\left( h_{2}\right) +h_{2}f\left( g\right) ] \\ &= h_{1}gf\left( g^{-1}\right) +h_{1}f\left( h_{2}\right) + h_{1}h_{2}f\left( g\right), \\ \left( g\cdot f\right) \left( h_{1}\right) & =gf\left( g^{-1}\right) +f\left( h_{1}\right) +h_{1}f\left( g\right) \end{align}$$ and $$ \begin{align} \left( g \cdot f\right) \left( h_{1}h_{2}\right) &=gf\left( g^{-1}\right) +f\left( h_{1}h_{2}\right) +h_{1}h_{2}f\left( g\right) \\ &= gf\left( g^{-1}\right) +h_{1}f\left( h_{2}\right) +f\left( h_{1}\right) +h_{1}h_{2}f\left( g\right). \end{align} $$
Note that $h_{1}[ f\left( g\right) +gf\left( g^{-1}\right)] = 0$, so the LHS is indeed equal to the RHS.