In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers. In this case, we say that the set is a generating set of or that generate. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. Examples The integers,, are a finitely generated abelian group. The integers modulo,, are a finite (hence finitely generated) abelian group. Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphism). In particular, the group of rational numbers is not finitely generated: if are rational numbers, pick a natural number coprime to all the denominators; then cannot be generated by. The group of non-zero rational numbers is also not finitely generated. The groups of
En álgebra abstracta, un grupo abeliano se llama generado finitamente si existen muchos elementos finitos de tal forma que cada uno de ellos puede ser escrito en la forma de algunos enteros. En este caso, decimos que el conjunto es un conjunto generador de o que generan. Cada grupo abeliano finito es generado finitamente. Los grupos abelianos generados finitamente pueden ser completamente clasificados. Ejemplos Los enteros, son un grupo abeliano generado finitamente. Los enteros modulo, son un grupo abeliano finito (por lo tanto generado finitamente). Cualquier suma directa de muchos grupos abelianos generados finitamente es nuevamente un grupo abeliano generado finitamente. Cada lattice forma un grupo abeliano libre generado finitamente. No hay otros ejemplos (para ser isomorfismo).