Remark

(1) Similarly it can be shown that condition (b) is equivalent to the condition (c') $H \neq \varnothing$ and the relations $x \in H$ and $y \in H$ imply $y^{-1}x \in H$. (2) For every subgroup H of G there are the following relations

For $H.H \subset H$ and $H^{-1} \subset H$ by (b). As $e \in H, H.H \supset e.H = H$ and taking inverses transforms the inclusion $H^{-1} \subset H$ into $H \subset H^{-1}$, whence formulae (1).

If H is a stable subgroup of G and K is a stable subgroup of H, clearly K is a stable subgroup of G.

The set $\{e\}$ is the smallest stable subgroup of G. The intersection of a family of stable subgroups of G is a stable subgroup. There is therefore a smallest stable subgroup H of G containing a given subset X of G; it is called the stable subgroup generated by X and X is called a generating system (or generating set) of H.