Well-definedness of a group action This is a fairly simply question but I want to check I'm not missing something important. 
If we have a group $G$ acting on a $G$-space $X$, then if we consider $X/N$ for some $G$-subspace $N$, is the restricted $G$-action $g.(x+N) = g.x +N$ still well-defined?
Sorry I may have used a term incorrectly, by G-subspace, I meant subspace of X invariant which is also a G space in its own right. 

It isn't a valid concept. $N$ is a subGROUP of $G$ not necciserily a subSPACE of $X$ and as such it is meaningless to talk about $x+N$.
If you mean 
$$(g+N).x=g.x$$
On the other hand, it depends on how the action is defined. Example where it doesnt work is $X=G=\Bbb Z$ and $N=2\Bbb Z$ with $g.x=g\cdot x$.
If you on the other hand let $N$ be a subSPACE the question of the well-definedness of 
$$g.(x+N)$$
is meaningful and in most cases, it will not be. HOWEVER if $N$ is invariant, that is for all $n\in N$ we have that $g.n=n$, with respect to the group action then it is well-defined.
