Finitely generated abelian group

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In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form

with integers n1,...,ns. In this case, we say that the set {x1,...,xs} is a generating set of G or that x1,...,xs generate G.

Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.

Contents

Examples

  • the integers (Z,+) are a finitely generated abelian group
  • the integers modulo n Zn are a finitely generated abelian group
  • any direct sum of finitely many finitely generated abelian groups is again finitely generated abelian

There are no other examples (up to isomorphism). The group (Q,+) of rational numbers is not finitely generated: if x1,...,xs are rational numbers, pick a natural number w coprime to all the denominators; then 1/w cannot be generated by x1,...,xs.

Classification

The fundamental theorem of finitely generated abelian groups (which is a special case of the structure theorem for finitely generated modules over a principal ideal domain) can be stated two ways (analogously with PIDs):

Primary decomposition

The primary decomposition formulation states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups. A primary cyclic group is one whose order is a power of a prime. That is, every finitely generated abelian group is isomorphic to a group of the form

where the rank n ≥ 0, and the numbers q1,...,qt are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q1,...,qt are (up to rearranging the indices) uniquely determined by G.

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