Let $G=S^{1}$ be a group with multiplication and $V=\mathbb{C}$ be a vector space of complex numbers. Let $G$ acts on $V$ as $(g,z)=gz$ usual multiplication of complex numbers. Then the orbit $Gz$ will be a circle centered at $0$ and having radius $|z|$. The orbit space $V/G= \{ Gz : z \in V \}$ is the set of concentric circles centered at origin. How can we define a vector space structure on $ V/G $ ? First we will try to define addition : $ Gz_{1}+Gz_{2} $
We cannot define it as $ G z_{1}+Gz_{2}= G(z_{1}+z_{2}) $ . Consider the case $ Gz_{1}=Gz_{2}=S^{1} $. Let us take $ 1 $ and $ -1 $ on $ G1=G(-1)= S^{1} $. Their sum will be $ G0 =0 $ but if we take other two points $ 1 $ and $ i $ on same orbit their sum will go to the orbit $ G(1+i) $. Hence this type of vector addition is not well defined.
Suppose we define it as $Gz_{1}+Gz_{2}= G(z_{1}.z_{2})$. Then multiplication of two points on two concentric circles with radius $r_{1}$ and $r_{2}$ will give a point on a concentric circle with radius $r_{1}.r_{2}$ rotated by an angle $\theta_{1}+\theta_{2}$. This operation is well defined. It is closed, commutative, and associative. $S^{1}=G{1}$ is the additive identity. But all the orbits except $G0=0$ has inverse. We cannot find multiplicative inverse to $0$.
Is their any other way to define vector space structure on orbit space in this case ?
Can we define vector space structure on orbit space $V/G$ in general (Let $G$ is a group and $V$ is a vector space ) ?
Do we require any other conditions for defining vector space structure on $V/G$ ?