Abelian groups

Relation to other mathematical topics

• 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.