Intersection of subgroups is a subgroup

Proposition (Intersection of subgroups). Let $G$ be a group and let $\{H_i\}_{i\in I}$ be a family of subgroups of $G$ . Then

H=\{g\in G:g\in H_i\text{ for every }i\in I\}

is a subgroup of $G$ . Thus $H=\bigcap_{i\in I}H_i$ , with the empty intersection understood to be $G$ .

Consequence

The intersection of all subgroups containing a subset $S\subseteq G$ is the smallest subgroup containing $S$ , called the subgroup generated by $S$ .