Exercise 24, § 7

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

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