Can the quotient set of a permutation equivalence relation be a vector space again?

Question

Let $x,y\in\mathbb{R}^N$ and consider the equivalence relation $x\sim y\Leftrightarrow \exists \sigma\in\operatorname{Sym}(\mathbb{R}^N)\,\, x = \sigma(y)$ for some permutation $\sigma$. Let $A = \mathbb{R}^N/\sim$ be the quotient set. Is it possible to define an addition between elements of $A$ (+ inverse and scalar multiplication) such that $A$ becomes a vector space?

Answer 1

Consider the vector $u\in\mathbb{R}^N$ with all entries equal to $1$. Then the map $a\mapsto [au]$ is an injection $\mathbb{R}\to\mathbb{R}^N$. By Cantor-Schröder-Bernstein, we conclude that $|\mathbb{R}^N|=|\mathbb{R}|$.

This is because $|\mathbb{R}^N|=|\mathbb{R}|$ (assuming $N$ is a positive integer), so $|\mathbb{R}^N/{\sim}|\le|\mathbb{R}^N|=|\mathbb{R}|$

Let $f\colon\mathbb{R}\to\mathbb{R}^N/{\sim}$ be a bijection. Use it to transfer the vector space structure.