---
tag: RQNAAJF6XY
kind: concept
name: Finitely generated abelian group
aliases: [classification-of-finitely-generated-abelian-groups, finitely-generated-abelian-groups, structure-theorem-for-finite-abelian-groups]
sources:
  - wikipedia: en:58527@r1353679405
    locator: Finitely generated abelian group
    note: adopted
    mention: wikipedia://en/58527
    sections: [Summary]
same_as:
  wikidata: Q142140
  wikipedia: Finitely generated abelian group
inventory: [Y70R7KWQN8]
---

## Summary

In abstract algebra, an [[abelian group|071DDFVG4J]] $(G,+)$ 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|0C4JS9RF5M]] $n_1,\dots, n_s$. In this case, we say that the set $\{x_1,\dots, x_s\}$ is a *[[generating set|0MBV2VHMXA]]* 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.