Set $G=\{C\subset\mathbb{R}^n\}$. Is there a binary operation $\cdot$ such that $(G,\cdot)$ is a group?

Question

Set $G=\{C\subset\mathbb{R}^n\}$. Is there a binary operation $\cdot$ such that $(G,\cdot)$ is a group?

The biggest trouble I'm having is concerning the existence of inverses. A standard notion of "set addition" is Minkowski addition, but this does not have inverses in general.

For those interested, I have asked a related question on this concerning the Hausdorff distance (also still unresolved).

Answer 1

Yes, consider the operation $\Delta$ of symmetric difference given by

$$A\Delta B=(A\cup B)\setminus(A\cap B).$$

Answer 2

Yes, the symmetric difference.

But even more abstractly, take any group $H$ of size $2^{2^{\aleph_0}}$ and a bijection $f : G \to H$. Then you can "transport" the structure from $H$ to $G$ so that $G$ becomes a group isomorphic to $H$.