Exercise 24, § 7

Let $G$ be a group, let $S = G - \{e\}$, and let $F = F(S)$. For every $s \in S$, denote by $x_s$ the corresponding element of $F$. If $s \in S, t \in S$, denote by $r_{s, t}$ the element of $F$ defined by:

Let $\varphi$ be the homomorphism of $F$ onto $G$ which maps $x_s$ onto $s$ and let $R$ be the kernel of $\varphi$. Now prove that the family $(r_{s, t})$, for $(s, t) \in S \times S$, is a basic family of the group $R$. (Apply Exer. 20 to the system of representatives of $G$ in $F$ formed by $e$ and the $x_s$.)