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I am given a group $G$ acting on $X$. Let $x$ be any element of $X$. Denote by $G\cdot x$ its orbit and $G_x$ its isotropy. I need to show orbit map $g\mapsto g\cdot x$ factors through the well defined bijective $G/G_x \to G\cdot x$ that maps $gG_x\to g\cdot x$, to define the following $G$- equivariant commutative diagram $\require{AMScd}$ \begin{CD} G\\ @VVV @.\\ G/G_x @>>> G\cdot x \end{CD} I am familiar with the notations of orbits and isotropy but I don't really know how will i be defining the elements of these groups to prove that it factors that way? Any hints?

mrtaurho
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Hannah_Zak
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    Do you know it is well-defined and bijective? Then the key is showing the map factors that way. – Thomas Andrews Aug 21 '21 at 14:42
  • @ThomasAndrews It is quite easy to show there is a one to one correspondence but after that, showing the map factors that way is giving a bit trouble here ? I don't get it how will I show that? what does it mean to factor in this case ? – Hannah_Zak Aug 21 '21 at 15:07

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For the map $\varphi\colon G\to G\cdot x$ to factor through $\overline\varphi\colon G/G_x\to G\cdot x$ means that there exists a map $\pi\colon G\to G/G_x$ factoring $\varphi$; that simply means $\varphi$ can be written as $\varphi=\overline\varphi\circ\pi$.

So what you have to do is finding the map $\pi$, for which there is an obvious canidate, and verify the equation $\varphi=\overline\varphi\circ\pi$ for your chosen map. Then the next step is verifing $G$-equivariance. Can you take it from here?

mrtaurho
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