# Quotient group, group action and quotient space

Let $$G$$ be a group acting faithfully on a topological space $$X$$, and $$N$$ a normal and abelian subgroup of $$G$$. Does this mean that one can define a faithful action of the quotient group $$G/N$$ on the quotient space $$X/N$$?

• Presumably, you want the faithful action of $G/N$ to be the induced action... – Arturo Magidin Jan 30 at 20:36
• Thank you. That’s helpful. This was probably not the right question to ask, and your comment suggests something more to understand. – user288227 Jan 30 at 21:39

Consider the discrete space $$X=\{1,2,3\}$$. $$S_3$$ acts faithfully on $$X$$ by definition. $$A_3 \subset S_3$$ is an abelian normal subgroup, but it acts transitively, so $$X/A_3$$ has only one point. Thus the quotient group $$S_3/A_3$$ cannot act faithfully on $$X/A_3$$.