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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

Contents

x = n1x1 + n2x2 + ... + nsxs

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.

Examples

  • The integers ( Z , + ) are a finitely generated abelian group.
  • The integers modulo n , ( Z n , + ) are a 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 ( Q , + ) of rational numbers is not finitely generated: if x 1 , … , x n are rational numbers, pick a natural number k coprime to all the denominators; then 1 / k cannot be generated by x 1 , … , x n . The group ( Q ∗ , ⋅ ) of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition (R, +) and real numbers under multiplication (R, ×) are also not finitely generated.

    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 principal ideal domains):

    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

    Z n ⊕ Z q 1 ⊕ ⋯ ⊕ Z q t ,

    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.

    Invariant factor decomposition

    We can also write any finitely generated abelian group G as a direct sum of the form

    Z n ⊕ Z k 1 ⊕ ⋯ ⊕ Z k u ,

    where k1 divides k2, which divides k3 and so on up to ku. Again, the rank n and the invariant factors k1, ..., ku are uniquely determined by G (here with a unique order).

    Equivalence

    These statements are equivalent because of the Chinese remainder theorem, which here states that Z m ≃ Z j ⊕ Z k if and only if j and k are coprime and m = jk.

    Corollaries

    Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of G. The rank of G is defined as the rank of the torsion-free part of G; this is just the number n in the above formulas.

    A corollary to the fundamental theorem is that every finitely generated torsion-free abelian group is free abelian. The finitely generated condition is essential here: Q is torsion-free but not free abelian.

    Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the group homomorphisms, form an abelian category which is a Serre subcategory of the category of abelian groups.

    Non-finitely generated abelian groups

    Note that not every abelian group of finite rank is finitely generated; the rank 1 group Q is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of Z 2 is another one.

    References

    Finitely generated abelian group Wikipedia
    (Text) CC BY-SA