Free abelian group
In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set together with an associative, commutative, and invertible binary operation. Conventionally, this operation is thought of as addition and its inverse is thought of as subtraction on the group elements. A basis is a subset of the elements such that every group element can be found by adding or subtracting a finite number of basis elements, and such that, for every group element, its expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis {1}. Addition of integers is commutative, associative, and has subtraction as its inverse operation, each integer can be formed by using addition or subtraction to combine some number of copies of the number 1, and each integer has a unique representation as an integer multiple of the number 1.
Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.
The elements of a free abelian group with basis B may be represented by expressions of the form where each coefficient ai is a nonzero integer, each factor bi is a distinct basis element, and the sum has finitely many terms. These expressions, and the group elements they represent, are also known as formal sums over B. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent the elements of a free abelian group is as the functions from B to the integers that have finitely many nonzero values, with pointwise addition of these functions as the group operation.
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