Let \(G\) be a group and let \(H\subseteq G\) be a subset. The subgroup generated by \(H\) is the intersection of all subgroups of \(G\) containing \(H\): \[ \gens{H} \coloneqq \bigcap \theset{M \suchthat H\subseteq M \leq G}. \]
Let \(G\) be a group and let \(H\subseteq G\) be a subset. Then \(\gens{H}\) is the smallest subgroup of \(G\) containing \(H\), and \[ \gens{H} = \theset{ h_1^{\varepsilon_1} \cdots h_n^{\varepsilon_n} \suchthat n\geq 0,\ h_i \in H,\ \varepsilon_i\in\theset{1,-1}}. \]