Proposition 2

Let X be a non-empty subset of a group with operators G and $\hat{X}$ the stable subset under the action of $\Omega$ on G generated by X. The stable subgroup generated by X is the stable subset under the law on G generated by the set $Y = \hat{X} \cup \hat{X}^{-1}$.

The latter subset Z is the set of compositions of finite sequences all of whose terms are elements of $\hat{X}$ or inverse of elements of $\hat{X}$: the inverse of such a composition is a composition of the same form (§ 2, no. 3, Corollary 1 to Proposition 5) and Z is stable under the action of $\Omega$, as is seen by applying § 3, no. 4, Proposition 1 to the homotheties of G, hence (Proposition 1) Z is a stable subgroup of G. Conversely, every stable subgroup containing X obviously contains Y and hence Z.