If I have a cyclic group $G = (a)$ acting on an abelian group $A$, I need to define a natural action of $G$ on the quotient space $A/B$, where $B$ is a normal subgroup of $A$ with the property that whenever $g\in G$ and $b\in B$, I have $g.b\in B$.
The only natural map that comes to mind is this:
Given a coset $x + B\in A/B$, I want to define $g.(x + B) = g.x + B$.
But I'm having difficulty showing that this action is well-defined. My problem is that the map $x\mapsto g.x$ is not linear in general.
If $x + B = y + B$, then $y - x\in B$ and so $g.(y-x)\in B$ by assumption. I want to use this to show that $g.y - g.x\in B$, which would confirm that $g.x + B = g.y + B$. But since $g.(y - x)\neq g.y - g.x$ I cannot make this jump.
Is this even the action that will work? or am I going down the wrong road?