Definition 4

Let G be a group with operators. A stable subgroup of G is a subset H of G with the following properties:

(i) $e \in H$; (ii) $x, y \in H \implies xy \in H$; (iii) $x \in H \implies x^{-1} \in H$; (iv) $x \in H$ and $\alpha \in \Omega$ imply $x^\alpha \in H$.

If H is a stable subgroup of G, the structure induced on H by the structure of a group with operators on G is the structure of a group with operators and the canonical injection of H into G is a homomorphism of groups with operators.

Let G be a group. A stable subgroup of G with the action of $\varnothing$ (no. 2), which is a subset of G satisfying conditions (i), (ii), (iii) of Definition 4, is called a subgroup of G. When we speak of a subgroup of a group of operators we shall always mean a subgroup of the underlying group of G. A subgroup of a group with operators G is not necessarily a stable subgroup of G.

Example (1). Let $\Sigma$ be a species of structure (Set Theory, IV, § 1, no. 4) and S a structure of species $\Sigma$ on a set E (loc. cit.). The set of automorphisms of S is a subgroup of $\mathscr{G}_E$.