Free abelian group

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free ${\displaystyle \mathbb {Z} }$-modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.

The elements of a free abelian group with basis ${\displaystyle B}$ may be described in several equivalent ways. These include formal sums over ${\displaystyle B}$, which are expressions of the form ${\textstyle \sum a_{i}b_{i}}$ where each ${\displaystyle a_{i}}$ is a nonzero integer, each ${\displaystyle b_{i}}$ is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of ${\displaystyle B}$, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from ${\displaystyle B}$ to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions.

Every set ${\displaystyle B}$ has a free abelian group with ${\displaystyle B}$ as its basis. This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free abelian group with basis ${\displaystyle B}$ may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of ${\displaystyle B}$. Alternatively, the free abelian group with basis ${\displaystyle B}$ may be described by a presentation with the elements of ${\displaystyle B}$ as its generators and with the commutators of pairs of members as its relators. The rank of a free abelian group is the cardinality of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups. The only free abelian groups that are free groups are the trivial group and the infinite cyclic group.