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 with an addition operation that is associative, commutative, and invertible. A basis is a subset such that every element of the group can be found by adding or subtracting basis elements, and such that every element's 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 is the sum or difference of some number of copies of the number 1, and each integer has a unique representation as an integer multiple of the number 1. 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 with an addition operation that is associative, commutative, and invertible. A basis is a subset such that every element of the group can be found by adding or subtracting basis elements, and such that every element's 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 is the sum or difference of 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 described in several equivalent ways.These include formal sums over B, which are expressions of the form ∑ a i b i {displaystyle sum a_{i}b_{i}} where each coefficient ai is a nonzero integer, each factor bi 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 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 B to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions. Every set B has a free abelian group with 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 group with basis B may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of B. Alternatively, the free abelian group with basis B may be described by a presentation with the elements of 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 integers, under the addition operation, form a free abelian group with the basis {1}. Every integer n is a linear combination of basis elements with integer coefficients: namely, n = n × 1, with the coefficient n. The two-dimensional integer lattice, consisting of the points in the plane with integer Cartesian coordinates, forms a free abelian group under vector addition with the basis {(0,1), (1,0)}. Letting these basis vectors be denoted   e 1 = ( 1 , 0 ) {displaystyle e_{1}=(1,0)} and   e 2 = ( 0 , 1 ) {displaystyle e_{2}=(0,1)} , the element (4,3) can be written In this basis, there is no other way to write (4,3). However, with a different basis such as {(1,0),(1,1)}, where   f 1 = ( 1 , 0 ) {displaystyle f_{1}=(1,0)} and   f 2 = ( 1 , 1 ) {displaystyle f_{2}=(1,1)} , it can be written as More generally, every lattice forms a finitely-generated free abelian group. The d-dimensional integer lattice has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if M is a d × d integer matrix with determinant ±1, then the rows of M form a basis, and conversely every basis of the integer lattice has this form. For more on the two-dimensional case, see fundamental pair of periods. The direct product of two free abelian groups is itself free abelian, with basis the disjoint union of the bases of the two groups. More generally the direct product of any finite number of free abelian groups is free abelian. The d-dimensional integer lattice, for instance, is isomorphic to the direct product of d copies of the integer group Z.