I'm reading the introductory book on L-theory, Algebraic and geometric surgery by Andrew Ranicki, available at 1.
Here is the statement in the page 254 Example 11.18.
If $G$ is a group without $2$-torsion, then (there is) a decomposition $ G=\lbrace 1\rbrace\cup S\cup S^{-1}$ which determines a $\mathbb{Z}[\mathbb{Z}/2]$-module splitting of $\mathbb{Z}[G]=\mathbb{Z}\oplus(\mathbb{Z}[S]\oplus\mathbb{Z}[S^{-1}])$.
Here, how can I decompose $G$ into $G=\lbrace 1\rbrace\cup S\cup S^{-1}$ for $G$ a group without $2$-torsion?