Given $G$ a group acting on $A$ an abelian group. Let $B$ be subgroup of $A$. Assume that $GB \subset B$. What is a natural way to define an action of $G$ on $A/B$ ?
An obvious idea: for $aB \in A/B $, $G$ acts by sending $aB$ to $(ga)B$. However, to check if this is well-defined, one need to check if $(ga_1)B = (ga_2)B$ whenever $a_1B =a_2B$. So far, we have $a_2^{-1}a_1 \in B$, thus $g(a_2^{-1}a_1)\in B$.
But how can we show $(ga_2)^{-1}ga_1 \in B$? Is it true that $g(xy) = g(x)g(y)$ where $x,y \in A$?
Any correction or suggestion is greatly appreciated. Thanks!