Definition of crossed homomorphisms 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 $N\trianglelefteq G$)
 A: 
Claim: $\left( g\cdot f\right) \left( h_{1}h_{2}\right) =h_{1}\left( g\cdot f\right) \left( h_{2}\right) +\left( g\cdot f\right) \left( h_{1}\right) $ for all $h_1 , h_2 \in H$.

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.
