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 $X/N$?

  • $\begingroup$ Presumably, you want the faithful action of $G/N$ to be the induced action... $\endgroup$ – Arturo Magidin Jan 30 at 20:36
  • $\begingroup$ Thank you. That’s helpful. This was probably not the right question to ask, and your comment suggests something more to understand. $\endgroup$ – 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 $X/A_3$.

  • $\begingroup$ Thank you for the answer! $\endgroup$ – user288227 Jan 30 at 21:40

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