Finitely generated abelian group

Summary

In abstract algebra, an abelian group $(G,+)$ is called finitely generated if there exist finitely many elements $x_1,\dots,x_s$ in $G$ such that every $x$ in $G$ can be written in the form $x = n_1x_1 + n_2x_2 + \cdots + n_sx_s$ for some integers $n_1,\dots, n_s$. In this case, we say that the set $\{x_1,\dots, x_s\}$ is a generating set of $G$ or that $x_1,\dots, x_s$ generate $G$. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups.