| A group $G$ is {\em finitely generated} if there is a finite subset $X\subseteq G$ such that $X$ generates $G$. That is, every element of $G$ is a product of elements of $X$ and inverses of elements of $X$. Or, equivalently, no proper subgroup of $G$ contains $X$. |
A group is finitely generated if there is a finite subset which generates it as a group (so that all elements of the group can be written as products of these elements). |
