Abelian Group - Relation To Other Mathematical Topics

Relation To Other Mathematical Topics

Many large abelian groups possess a natural topology, which turns them into topological groups.

The collection of all abelian groups, together with the homomorphisms between them, forms the category Ab, the prototype of an abelian category.

Nearly all well-known algebraic structures other than Boolean algebras, are undecidable. Hence it is surprising that Tarski's student Szmielew (1955) proved that the first order theory of abelian groups, unlike its nonabelian counterpart, is decidable. This decidability, plus the fundamental theorem of finite abelian groups described above, highlight some of the successes in abelian group theory, but there are still many areas of current research:

• Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood;
• There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
• While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature.
• Many mild extensions of the first order theory of abelian groups are known to be undecidable.
• Finite abelian groups remain a topic of research in computational group theory.

Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is:

• Undecidable in ZFC, the conventional axiomatic set theory from which nearly all of present day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
• Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom;
• Decidable if ZFC is augmented with the axiom of constructibility (see statements true in L).