Finitely generated abelian group
Encyclopedia
In abstract algebra
, an abelian group
(G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form
with integer
s n1,...,ns. In this case, we say that the set {x1,...,xs} is a generating set
of G or that x1,...,xs generate G.
Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.
There are no other examples (up to isomorphism). The group (Q,+) of rational number
s is not finitely generated: if x1,...,xs are rational numbers, pick a natural number
w coprime
to all the denominators; then 1/w cannot be generated by x1,...,xs. The group (Q*,*) of non-zero rational numbers is also not finitely generated.
(which is a special case of the structure theorem for finitely generated modules over a principal ideal domain
) can be stated two ways (analogously with PID
s):
s and infinite cyclic group
s. A primary cyclic group is one whose order is a power of a prime
. That is, every finitely generated abelian group is isomorphic to a group of the form
where the rank
n ≥ 0, and the numbers q1,...,qt are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q1,...,qt are (up to
rearranging the indices) uniquely determined by G.
where k1 divides
k2, which divides k3 and so on up to ku. Again, the rank n and the invariant factor
s k1,...,ku are uniquely determined by G (here with a unique order).
, which here states that Zm is isomorphic to the direct product of Zj and Zk if and only if j and k are coprime
and m = jk.
of finite rank
and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup
of G. The rank of G is defined as the rank of the torsion-free part of G; this is just the number n in the above formulas.
A corollary
to the fundamental theorem is that every finitely generated torsion-free abelian group is free abelian. The finitely generated condition is essential here: Q is torsion-free but not free abelian.
Every subgroup
and factor group of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the group homomorphism
s, form an abelian category
which is a Serre subcategory
of the category of abelian groups
.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
(G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form
- x = n1x1 + n2x2 + ... + nsxs
with integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s n1,...,ns. In this case, we say that the set {x1,...,xs} is a generating set
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
of G or that x1,...,xs generate G.
Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.
Examples
- the integers (Z,+) are a finitely generated abelian group
- the integers modulo nModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
Zn are a finitely generated abelian group - any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group
There are no other examples (up to isomorphism). The group (Q,+) of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s is not finitely generated: if x1,...,xs are rational numbers, pick a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
w coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
to all the denominators; then 1/w cannot be generated by x1,...,xs. The group (Q*,*) of non-zero rational numbers is also not finitely generated.
Classification
The fundamental theorem of finitely generated abelian groups(which is a special case of the structure theorem for finitely generated modules over a principal ideal domain
Structure theorem for finitely generated modules over a principal ideal domain
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in...
) can be stated two ways (analogously with PID
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
s):
Primary decomposition
The primary decomposition formulation states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groupPrimary cyclic group
In mathematics, a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p.That is, it has the formfor some prime number p, and natural number m....
s and infinite cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
s. A primary cyclic group is one whose order is a power of a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. That is, every finitely generated abelian group is isomorphic to a group of the form
where the rank
Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
n ≥ 0, and the numbers q1,...,qt are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q1,...,qt are (up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
rearranging the indices) uniquely determined by G.
Invariant factor decomposition
We can also write any finitely generated abelian group G as a direct sum of the formwhere k1 divides
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
k2, which divides k3 and so on up to ku. Again, the rank n and the invariant factor
Invariant factor
The invariant factors of a module over a principal ideal domain occur in one form of the structure theorem for finitely generated modules over a principal ideal domain.If R is a PID and M a finitely generated R-module, then...
s k1,...,ku are uniquely determined by G (here with a unique order).
Equivalence
These statements are equivalent because of the Chinese remainder theoremChinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
, which here states that Zm is isomorphic to the direct product of Zj and Zk if and only if j and k are coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
and m = jk.
Corollaries
Stated differently the fundamental theorem says that a finitely-generated abelian group is the direct sum of a free abelian groupFree abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
of finite rank
Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup
Torsion subgroup
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...
of G. The rank of G is defined as the rank of the torsion-free part of G; this is just the number n in the above formulas.
A corollary
Corollary
A corollary is a statement that follows readily from a previous statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective...
to the fundamental theorem is that every finitely generated torsion-free abelian group is free abelian. The finitely generated condition is essential here: Q is torsion-free but not free abelian.
Every subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
and factor group of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s, form an abelian category
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
which is a Serre subcategory
Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and...
of the category of abelian groups
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....
.