> [!NOTE] Theorem (Necessary condition for subgroup) > Let $(G,\cdot)$ be a [[Groups|group]]. Suppose $H\subset G$ then $(H,\cdot)$ is a [[Subgroup|subgroup]] iff > 1. $H\neq \emptyset$ > 2. $\forall a,b \in H \;(a\cdot b^{-1}\in H)$ *Proof*. ...