Groups Generated by Sets

If we extend the definition of a cyclic group to include more generators, e.g., $⟨g_1,g_2⟩$ would create a group the exact same. Formally, we can see that…

Let $G$ be a group and let $S\subseteq G$. Then the group generated by $S$ is $⟨S⟩=\{a_1^{n_1}{a}_2^{n_2}\cdots a_m^{n_m}:m\in \mathbb{N}\text{ and }{n}_i\in \mathbb{Z}\text{ and }{a}_i\in S\text{ for all }1\le i\le m\}$. Or, similarly to before, the smallest subgroup of $G$ that contains all elements in $S$.