Finitely-Presented Abelian Groups

Abelian groups are of interest not only for their intrinsic interest but also because many of the important groups arising in number theory and topology are abelian.

• Construction as a quotient of a free abelian group
• Direct product, free product
• Arithmetic
• Construction of subgroups and quotient groups
• Elementary divisors, primary invariants
• Factor basis, divisor basis, primary basis
• Torsion subgroup, torsion-free subgroup, p-primary component
• Homomorphisms: Image, kernel, cokernel
• Composition series, maximal subgroups, subgroup lattice (of a finite group)
• Character table of a finite group
• The group of homomorphisms Hom(A,B), where A and B are finite abelian groups
• Abelian quotient of any group (with its natural homomorphism)
• Conversion between $\mbox{\bf Z}$-modules and abelian groups
• Functors from rings and fields onto abelian groups

The Hermite and Smith normal form algorithms are used to construct a normal form for subgroups and quotient groups of abelian groups.