This question may be very trivial in this area, but I am a beginner to this and I stuck here. Could anyone please help me here!
Let $\Gamma$ be a group whose identity is $e$.
Let $X$ be a set and $∗:\Gamma×S\rightarrow S$ be a group action.
Let $[x]$ is the orbit (equivalence class) of $x\in X$ under the group action $\Gamma$.
Now I want to define addition in the quotient space $X/\Gamma$.
Here is my attempt:
Take $x'\in [x]$ and $y'\in [y]$ where $[x]\cap[y]=\phi$.
Then to define sum uniquely, I need to show $x+x'\sim_G y+y'$.
To show the above: Since $x'\in [x]$ and $y'\in [y]$ $$\exists g_1\in \Gamma: x'=g_1*x, \;\;\exists g_2\in \Gamma: y'=g_2*y.$$
Now $$x'+y'=g_1*x+g_2*y$$ Here I stuck: since $g_1$ and $g_2$ may be different element of $\Gamma$, then how to show $x'+y'\sim_\Gamma x+y$?
When you have an action of a group to a set then the quotient doesn't always form a group. Take for example the natural action of the group of orthogonal matrices $O(n)$ on the sphere $\mathbb{S}^{n-1}$. Let $p$ be the north pole, i.e. $p=(0,0,\cdots,0,1)$, then you have that $$O(n)/\mathrm{Stab}(p)\cong \mathbb{S}^{n-1},$$ where $\cong$ means diffeomorphism and $\mathrm{Stab}(p)$ is the stabilizer of $p$ in $O(n)$ (which is isomophic to $O(n-1)$). Thus, if the quotient $O(n)/\mathrm{Stab}(p)$ could form a group then we could pass that group structure to $\mathbb{S}^{n-1}$ via the diffeomorphism. But this is a contradiction since (due to J.Milnor) we know that the only spheres that admit a group structure are $\mathbb{S}^1$ and $\mathbb{S}^3$.