Abelian group

In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that the group operation * is commutative, so that for all a and b in G, a * b = b * a. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, commutativity is the ability to change the order of something without changing the end result Abelian groups are named after Norwegian mathematician Niels Henrik Abel. Norway ( Norwegian: Norge ( Bokmål) or Noreg ( Nynorsk) officially the Kingdom of Norway, is a Constitutional A mathematician is a person whose primary area of study and research is the field of Mathematics. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation Groups in which the group operation is not commutative are called non-abelian (or non-commutative). Since the group operation in an abelian group is commutative as well as associative, the value of a product of group elements is independent of the order in which the product is calculated. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, although infinite abelian groups are the subject of current research.

1 Notation
2 Examples
3 Multiplication table
4 Properties
5 Finite abelian groups
- 5.1 Automorphisms of finite abelian groups
6 Relation to other mathematical topics
7 A note on the typography
8 See also
9 References

Notation

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

Convention	Operation	Identity	Powers	Inverse
Addition	x + y	0	nx	−x
Multiplication	x * y or xy	e or 1	xⁿ	x ⁻¹

The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. 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 When studying abelian groups apart from other groups, the additive notation is usually used.

Examples

Every cyclic group G is abelian, because if x, y are in G, then xy = a^maⁿ = a^{m + n} = a^{n + m} = aⁿa^m = yx. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Thus the integers, Z, form an abelian group under addition, as do the integers modulo n, Z/nZ. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

Every ring is an abelian group with respect to its addition operation. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication. In Mathematics, the real numbers may be described informally in several different ways

Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G Subgroups, quotients, and direct sums of abelian groups are again abelian. In Mathematics, a group G is called the direct sum of a set of Subgroups { H i } if each H

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

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. 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 If the group is G = {g₁ = e, g₂, . . . , g_n} under the operation ⋅, the (i, j)'th entry of this table contains the product g_i ⋅ g_j. 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 Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T}

This is true since if the group is abelian, then g_i ⋅ g_j = g_j ⋅ g_i. 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 + . In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an . . + x (n summands) and (−n)x = −(nx). In this way, G becomes a module over the ring Z of integers. 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 In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In fact, the modules over Z can be identified with the abelian groups.

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. In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i A typical example is the classification of finitely generated abelian groups. In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1

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. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function (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.

Somewhat akin to the dimension of vector spaces, every abelian group has a rank. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it 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 over the It is defined as the cardinality of the largest set of linearly independent elements of the group. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors The integers and the rational numbers have rank one, as well as every subgroup of the rationals. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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 Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1

$\mathbb{Z}_{mn}$ is isomorphic to the direct product of $\mathbb{Z}_m$ and $\mathbb{Z}_n$ 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

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,. . . ,k_u are powers of primes
where k₁ divides k₂, which divides k₃ and so on up to k_u. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without

For example, $\mathbb{Z}/15\mathbb{Z}\cong\mathbb{Z}_{15}$ can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: $\mathbb{Z}_{15}\cong\{0, 5, 10\}\oplus\{0, 3, 6, 9, 12\}$ . 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

For another example, every abelian group of order 8 is isomorphic to either $\mathbb{Z}_8$ (the integers 0 to 7 under addition modulo 8), $\mathbb{Z}_4\oplus\mathbb{Z}_2$ (the odd integers 1 to 15 under multiplication modulo 16), or $\mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{Z}_2$ .

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. 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 To do this, one uses the fact (which will not be proved here) that if G splits as a direct sum H $\oplus$ K of subgroups of coprime order, then Aut(H $\oplus$ K) $\cong$ Aut(H) $\oplus$ Aut(K). In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than

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). In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which Fix a prime p and suppose the exponents e_i of the cyclic factors of the Sylow p-subgroup are arranged in increasing order:

$e_1\leq e_2 \leq\cdots\leq e_n$

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

$\mathbb{Z}_{p^{e_1}} \oplus \cdots \oplus \mathbb{Z}_{p^{e_n}}$

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. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Another special case is when n is arbitrary but e_i = 1 for 1 ≤ i ≤ n. Here, one is considering P to be of the form

$\mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p$ ,

so elements of this subgroup can be viewed as comprising a vector space of dimension n over the finite field of p elements $\mathbb{F}_p$ . The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

$\mathrm{Aut}(P)\cong\mathrm{GL}(n,\mathbb{F}_p)$ ,

which is easily shown to have order

$|\mathrm{Aut}(P)|=(p^n-1)\cdots(p^n-p^{n-1})$ .

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

$d_k=\mathrm{max}\{r|e_r = e_k^{\,}\}$

and

$c_k=\mathrm{min}\{r|e_r=e_k^{\,}\}$

then one has in particular d_k ≥ k, c_k ≤ k, 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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the

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 In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist

Nearly all well-known algebraic structures other than Boolean algebra, are undecidable. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Logic, the term decidable refers to the existence of an Effective method for determining membership in a set of formulas 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, highlight some of the successes in abelian group theory, but there are still many areas of current research:

Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood;
There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the cased of countable mixed groups is much less mature. Infinitely generated Abelian groups have very complex structure and are far less well understood than Finitely generated abelian groups.
Many mild extensions of the first order theory of abelian groups are known to be undecidable.
Finite abelian groups remain a topic of research in computational group theory.

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:

Undecidable in ZFC, the conventional axiomatic set theory from which nearly all of present day mathematics can be derived. In Group theory, a branch of Abstract algebra, the Whitehead problem is the following question Is every Abelian group A with 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 Saharon Shelah (שהרן שלח 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 The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC;
Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom;
Decidable if ZFC is augmented with the axiom of constructibility (see statements true in L). Undecidable has more than one meaning;In Mathematical logic: A Decision problem is called (recursively undecidable if no Algorithm can Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite Gödel universe redirects here For Kurt Gödel 's cosmological solution to the Einstein field equations, see Gödel metric. Here is a list of propositions that hold in the Constructible universe (denoted L The generalized continuum hypothesis and as a consequence

A note on the typography

Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is spelled with a lowercase a, rather than an uppercase A, indicating how ubiquitous the concept is in modern mathematics. In Grammar, an adjective is a word whose main syntactic role is to modify a Noun or Pronoun, giving more information about the "A proper name a word that answers the purpose of showing what thing it is that we are talking about" writes John Stuart Mill in A System of Logic A mathematician is a person whose primary area of study and research is the field of Mathematics. ^[1]

References

^ Abel Prize Awarded: The Mathematicians' Nobel

Fuchs, László (1970) Infinite abelian groups, Vol. In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup In Mathematics, class field theory is a major branch of Algebraic number theory. In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup In Group theory an elementary abelian group is a finite Abelian group, where every nontrivial element has order p where p is a prime In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 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 In Mathematics, in particular in Harmonic analysis and the theory of Topological groups Pontryagin duality explains the general properties of the Fourier Infinitely generated Abelian groups have very complex structure and are far less well understood than Finitely generated abelian groups. I. Pure and Applied Mathematics, Vol. 36. New York-London: Academic Press. xi+290 pp. MR 0255673
------ (1973) Infinite abelian groups, Vol. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many II. Pure and Applied Mathematics. Vol. 36-II. New York-London: Academic Press. ix+363 pp. MR 0349869
Hillar, Christopher and Rhea, Darren, Automorphisms of finite abelian groups. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Amer. Math. Monthly 114 (2007), no. 10, 917-923. [1].
Szmielew, Wanda (1955) "Elementary properties of abelian groups," Fundamenta Mathematica 41: 203-71.

Dictionary

-noun

(algebra) a group in which the group operation is commutative

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents