> [!NOTE] Definition (Subgroup) > Let $(G,\cdot)$ be a [[Groups|group]]. Let $H$ be a [[Subsets|subset]] and suppose that $(H,\cdot)$ is also a group. Then $(H,\cdot)$ is a *subgroup* of $(G,\cdot)$ # Properties **Subgroup tests:** Let $G$ be a group. (**Two-step subgroup test**) If $H \subset G$ then a [[Two-Step Subgroup Test|necessary condition]] for $H$ being a subgroup is (1) the identity of the binary operation lies in $H$; (2) $a,b\in H\implies a\cdot b\in H$; and (3) $a\in H \implies a^{-1}\in H.$ (**One-step test**) a more succinct [[One-Step Subgroup Test|necessary condition]] is that $H$ is non empty and $a,b\in H \implies a\cdot b^{-1}\in H.$ **New subgroups from old**: By [[Intersection of Subgroups is Subgroup]], ... **Congruence modulo subgroup**: [[Coset]]; [[Congruence Modulo Subgroup]]; [[Coset space]]; If [[Normal Subgroup]]. # Applications **Examples**: Let $g$ be a group element. Then $\langle g \rangle =\{ g^{n} \mid n\in \mathbb{Z} \}$ is a subgroup called the [[Generated Subgroup|subgroup generated]] by $g.$