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Encyclopedia > Abelian group

In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that "*" commutes: for all a and b in G, a * b = b * a. The name honors Niels Henrik Abel. Groups that are not commutative are called non-abelian (or non-commutative). Since "*" commutes as well as associates, objects linked by "*" may be permuted at will while preserving truth. The theory of abelian groups is generally easier than that of their non-abelian counterparts, although infinite abelian groups are the subject of current research. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This picture illustrates how the hours on a clock form a group under modular addition. ... A map or binary operation from a set to a set is said to be commutative if, (A common example in school-math is the + function: , thus the + function is commutative) Otherwise, the operation is noncommutative. ... Niels Henrik Abel (August 5, 1802â€“April 6, 1829), Norwegian mathematician, was born in Nedstrand, near FinnÃ¸y where his father acted as rector. ...

There are two main notational conventions for abelian groups — additive and multiplicative.

Convention Operation Identity Powers Inverse Direct sum/product
Addition x + y 0 nx x GH
Multiplication x * y or xy e or 1 xn x −1 G × H

The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. When studying abelian groups apart from other groups, the additive notation is usually used. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

## Examples

Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form an abelian group under addition, as do the integers modulo n, Z/nZ. 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 a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... The integers are commonly denoted by the above symbol. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ...

Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication. In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. In group theory, given a group G under a binary operation *, we say that some subset H of G is 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 is a group... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In group theory, a group G is called the direct sum of a set of subgroups {Hi} if each Hi is a normal subgroup of G each distinct pair of subgroups has trivial intersection, and G = <{Hi}>; in other words, G is generated by the subgroups {Hi}. If G is...

Matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

## Multiplication table

To verify that a finite group is abelian, a table (matrix) - known as a Cayley table - can be constructed in a similar fashion to a multiplication table. If the group is G = {g1 = e, g2, ..., gn} under the operation ⋅, the (i, j)'th entry of this table contains the product gigj. The group is abelian if and only if this table is symmetric about the main diagonal (i.e. if the matrix is a symmetric matrix). In mathematics, a finite group is a group which has finitely many elements. ... A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the groups elements in a square table reminiscent of an addition or multiplication table. ... In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. ... â†” â‡” â‰¡ logical symbols representing iff. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...

This is true since if the group is abelian, then gigj = gjgi. This implies that the (i, j)'th entry of the table equals the (j, i)'th entry - i.e. the table is symmetric about the main diagonal.

## Properties

If n is a natural number and x is an element of an abelian group G written additively, then nx can be defined as x + x + ... + x (n summands) and (−n)x = −(nx). In this way, G becomes a module over the ring Z of integers. In fact, the modules over Z can be identified with the abelian groups. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

Theorems about abelian groups (i.e. modules over the principal ideal domain Z) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups. Look up module in Wiktionary, the free dictionary. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... 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 x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...

If f, g : G  →  H are two group homomorphisms between abelian groups, then their sum f + g, defined by (f + g)(x) = f(x) + g(x), is again a homomorphism. (This is not true if H is a non-abelian group.) The set Hom(G, H) of all group homomorphisms from G to H thus turns into an abelian group in its own right. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the cardinality of the largest set of linearly independent elements of the group. The integers and the rational numbers have rank one, as well as every subgroup of the rationals. 2-dimensional renderings (ie. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

## Finite abelian groups

The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0. In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... 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 x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...

Zmn is isomorphic to the direct product of Zm and Zn if and only if m and n are coprime. In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and âˆ’1, or equivalently, if their greatest common divisor is 1. ...

Therefore we can write any finite abelian group G as a direct product of the form

$mathbb{Z}_{k_1} oplus cdots oplus mathbb{Z}_{k_u}$

in two unique ways:

• where the numbers k1,...,ku are powers of primes
• where k1 divides k2, which divides k3 and so on up to ku.

Thus we have 3 2 or 6, 5 2 or 10, 4 3 or 12, 3 2 2 or 6 2, 7 2 or 14, and 5 3 or 15, but anyway 2 2, 4 2, 2 2 2, 3 3, 8 2, 4 4, 4 2 2, and 2 2 2 2. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

For example, Z/15Z = Z/15 can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: Z/15 = {0, 5, 10} ⊕ {0, 3, 6, 9, 12}. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

For another example, every abelian group of order 8 is isomorphic to either Z/8 (the integers 0 to 7 under addition modulo 8), Z/4  ⊕ Z/2 (the odd integers 1 to 15 under multiplication modulo 16), or Z/2  ⊕  Z/2  ⊕  Z/2.

See also list of small groups for finite abelian groups of order 16 or less. The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

### Automorphisms of finite abelian groups

One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group G. To do this, one uses the fact (which will not be proved here) that if G splits as a direct sum HK of subgroups of coprime order, then Aut(HK) ≅ Aut(H) ⊕ Aut(K). In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and âˆ’1, or equivalently, if their greatest common divisor is 1. ...

Given this, the fundamental theorem shows that to compute the automorphism group of G it suffices to compute the automorphism groups of the Sylow p-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of p). Fix a prime p and suppose the exponents ei of the cyclic factors of the Sylow p-subgroup are arranged in increasing order: The Sylow theorems of group theory, named after Ludwig Sylow, form a partial converse to Lagranges theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the...

e1e2 ≤ … ≤ en

for some n > 0. One needs to find the automorphisms of

Zpe1 ⊕ ... ⊕ Zpen

One special case is when n = 1, so that there is only one cyclic prime-power factor in the Sylow p-subgroup P. In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when n is arbitrary but ei = 1 for 1 ≤ in. Here, one is considering P to be of the form 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 a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

Zp ⊕ … ⊕ Zp,

so elements of this subgroup can be viewed as comprising a vector space of dimension n over the finite field of p elements Fp. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

Aut(P) ≅ GL(n, Fp),

which is easily shown to have order

|Aut(P)| = (pn − 1)…(pnpn−1).

In the most general case, where the ei and n are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines

dk = max{r | er = ek}

and

ck = min{r | er = ek}

then one has in particular dkk, ckk, and

$|mathrm{Aut}(P)| = left(prod_{k=1}^n{p^{d_k} - p^{k-1}}right)left(prod_{j=1}^n{(p^{e_j})^{n-d_j}}right)left(prod_{i=1}^n{(p^{e_i-1})^{n-c_i+1}}right).$

One can check that this yields the orders in the previous examples as special cases (see [Hillar,Rhea]).

## Relation to other mathematical topics

Many large abelian groups possess a natural topology, which turns them into topological groups. A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G Ã— G â†’ G and the inverse operation G â†’ G are continuous maps. ...

The collection of all abelian groups, together with the homomorphisms between them, forms the category Ab, the prototype of an abelian category. In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ... 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 nice properties. ...

Nearly all well-known algebraic structures other than Boolean algebra, are undecidable. Hence it is surprising that Tarski's student Szmielew (1955) proved that the first order theory of abelian groups, unlike its nonabelian counterpart, is decidable. This decidability, plus the fundamental theorem of finite abelian groups described above, suggests that abelian group theory is no longer very interesting mathematically. This conclusion would be mistaken, because: In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... The word decidable has formal meaning in computability theory, the theory of formal languages, and mathematical logic. ... The word decidable has formal meaning in computability theory, the theory of formal languages, and mathematical logic. ...

• Only rank one torsion-free abelian groups are well understood;
• There are many unsolved problems in the theory of infinite-rank abelian groups;
• Many mild extensions of the first order theory of abelian groups are known to be undecidable.

Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is: In abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In group theory, the Whitehead problem is the following question: Is every abelian group A with Ext1(A, Z) = 0 a free abelian group? The condition Ext1(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B &#8594; A is a surjective... 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. ... Saharon Shelah (, born July 3, 1945 in Jerusalem) is an Israeli mathematician. ...

The following is a list of mathematical statements that are undecidable in ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. ... This article or section is in need of attention from an expert on the subject. ... Undecidable has more than one meaning: In mathematical logic: A decision problem is undecidable if there is no known algorithm that decides it. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, the constructible universe (or Gödels constructible universe) is a particular class of sets which can be described entirely in terms of simpler sets. ... The axiom of constructibility is a powerful statement that resolves many propositions in set theory and some interesting questions in analysis. ...

## A note on the typography

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, class field theory is a major branch of algebraic number theory. ... In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G1 generated by all the commutators of elements of G; that is, G1 = <[g,h] : g,h in G>. Note that the set of all commutators of the group is, generally, not a group (in...

## References

• Fuchs, László (1970) Infinite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New York-London: Academic Press. xi+290 pp. MR0255673
• ------ (1973) Infinite abelian groups, Vol. II. Pure and Applied Mathematics. Vol. 36-II. New York-London: Academic Press. ix+363 pp. MR0349869
• Hillar, Christopher. and Rhea, Darren Automorphisms of Finite Abelian Groups.
• Szmielew, Wanda (1955) "Elementary properties of abelian groups," Fundamenta Mathematica 41: 203-71.

Results from FactBites:

 NationMaster - Encyclopedia: Free abelian group (752 words) A typical example of a free abelian group is the direct sum Z ⊕ Z of two copies of the infinite cyclic group Z; a basis is {(1,0),(0,1)}. Note a point on terminology: a free abelian group is not the same as a free group that is abelian; in fact most free groups are not abelian. All free abelian groups are torsion free, and all finitely generated torsion free abelian groups are free abelian.
 Abelian group - Wikipedia, the free encyclopedia (963 words) The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. A typical example is the classification of finitely generated abelian groups. This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.
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