Let $(G,\ast)$ be a group. For every subset $A\subseteq G$ define $$\langle A\rangle:=\bigcap_{A\subset S\subset G}S$$ as the set intersection of all subgroups $S$ of $G$ containing $A.$
Obviously, $A\subset \langle A\rangle$ and $\langle A\rangle \subset S$ for every subgroup $S$ of $G.$ Therefore, $\langle A\rangle$ is the smallest subgroup of $G$ containing $A.$ We call $A$ the generating set of $\langle A\rangle$ and $\langle A\rangle$ the group generated by $A.$ If $A$ is finite, we write $\langle a_1,\ldots,a_n\rangle.$
Definitions: 1
Proofs: 2 3 4
Propositions: 5