# Definitions > [!NOTE] Definition 1 (Subgroup generated by $S$ is the set of all words over $S$) > Let $G$ be a [[Groups|group]]. Let $S\subset G$ be a subset. Then the subgroup generated by $S$ is given by $\langle S \rangle = \{ s_{1}^{\varepsilon_{1}}s_{2}^{\varepsilon_{2}}\cdots s_{m}^{\varepsilon_{m}} \mid m\in \mathbb{N}, s_{i}\in S, \varepsilon_{i}\in \{ 1,-1 \} \}$where $\mathbb{N}=\{ 0,1,2,3,\dots \}$ is the set of natural numbers. **Remark**: $s_{1}^{\varepsilon_{1}}s_{2}^{\varepsilon_{2}}\cdots s_{m}^{\varepsilon_{m}}$ is called a word in $S,$ that is, a finite product of elements of $S$ in any order. So $\langle S \rangle$ is the set of all words in $S.$ > [!NOTE] Definition 2 (Subgroup generated by $S$ is the intersection of subgroups containing $S$) > Let $G$ be a [[Groups|group]]. Let $S\subset G$ be a subset. Then the subgroup generated by $S$ is given by $ \langle S \rangle \bigcup_{S \subset M \leq G} M,$ that is, the intersection of all [[Subgroup|subgroups]] of $G$ containing $S.$ > [!NOTE] Definition 3 (Subgroup generated is the smallest subgroup containing $S$) > Let $G$ be a [[Groups|group]]. Let $S\subset G$ be a subset. Then the subgroup generated by $S$ is the smallest subgroup of $G$ containing $S.$ > [!NOTE] Definition (Subgroup generated by single element) > Let $G$ be a [[Groups|group]] and $g\in G.$ Then the *subgroup generated* by $g$ is the set of all [[Integer Power of Group Element|integer powers]] of $g$ denoted $\langle g \rangle = \{ g^{n} \mid n\in \mathbb{Z} \}$ # Properties By [[Subgroup Generated by Single Element is Indeed a Subgroup]], $\langle g \rangle$ is indeed a subgroup of $G.$ Moreover by [[Order of Group Element Equals Order of Subgroup Generated by Element]], the order of the subgroup is the same as the order of the group. More generally, it follows from [[Generated Subgroup is Subgroup]] that $\langle S \rangle \leq G$ for all subsets $S \subset G.$ In particular, the given definitions are equivalent. # Applications **Finitely generated groups**: A group is [[Cyclic Group|cyclic]] if it is equal to the subgroup by one of its members. More generally, a group is [[Finitely Generated Group|finitely generated]] if it equals the subgroup generated by some finite subset.