I'm reading the book "The Banach-Tarski Paradox" by Stan Wagon. The Theorem 1.5 (page no:7) it states the following:
Theorem 1.5 (AC). $S^1$ is countably $SO_2$-paradoxical. If $G$ denotes the group of translations modulo $1$ acting on $[0,1)$, then $[0,1)$ is countably $G$-paradoxical.
But I can't understand what is meant here by **"$G$ denotes the group of translations modulo $1$". What group is this?
Please give its definition. Thank you.