Proposition 1

Let G be a group with operators and H a subset of G which is stable under the homotheties of G. The following conditions are equivalent:

(a) H is a stable subgroup of G. (b) H is non-empty and the relations $x \in H, y \in H \implies xy \in H$ and $x^{-1} \in H$. (c) H is non-empty and the relations $x \in H, y \in H \implies xy^{-1} \in H$. (d) H is stable under the law on G and the law of composition induced on H by the law of composition on G is a group law.

Clearly (a) implies (b). We show that (b) implies (a). It suffices to show that H contains the identity element of G. As the subset H is non-empty, let $x \in H$. Then $x^{-1} \in H$ and $e = xx^{-1} \in H$. Clearly (b) implies (c). We show that (c) implies (b). First of all since H is non-empty it contains an element x. Hence xx^{-1} = e is an element of H. For every element x of H, $x^{-1} = ex^{-1}$ belongs to H; hence the relations $x \in H, y \in H$ imply $x(y^{-1})^{-1} = xy \in H$. Clearly (a) implies (d). We show that (d) implies (a): the canonical injection of H into G is a group homomorphism; hence $e \in H$ and the relation $x \in H$ implies $x^{-1} \in H$ (no. 1).