Subgroup generated by a subset

Remarks

By the intersection property, $\langle S \rangle$ is a subgroup, and it is contained in each subgroup that contains $S$. Its elements are the products $s_1^{\varepsilon_1} s_2^{\varepsilon_2} \cdots s_k^{\varepsilon_k}$ with $k \geq 0$, $s_i \in S$ and $\varepsilon_i \in \{1, -1\}$, where the empty product ($k = 0$) is the identity. In particular, $\langle \emptyset \rangle = \{e\}$.