Vad är abelian group med exempel

Abelian group

Commutative group (mathematics)

For the group described bygd the archaic use of the related begrepp "Abelian linear group", see Symplectic group.

In mathematics, an abelian group, also called a commutative group, fryst vatten a group in which the result of applying the group operation to two group elements does not depend on the beställning in which they are written.

That fryst vatten, the group operation fryst vatten commutative. With addition as an operation, the integers and the real numbers struktur abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after Niels Henrik Abel.^[1]

The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras.

The theory of abelian groups fryst vatten generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.

In an abelian group, the equation 'a + b = b + a' holds for all elements a and b in the group.

Definition

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An abelian group fryst vatten a set, tillsammans with an operation that combines any two elements and of to form eller gestalt another element of denoted . The emblem fryst vatten a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, , must satisfy fyra requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation fryst vatten defined for any ordered pair of elements of A, that the result fryst vatten well-defined, and that the result belongs toA):

Associativity: For all , , and in , the equation holds.
Identity element: There exists an element in , such that for all elements in , the equation holds.
Inverse element: For each in there exists an element in such that , where fryst vatten the identity element.
Commutativity: For all , in , .

A group in which the group operation fryst vatten not commutative fryst vatten called a "non-abelian group" or "non-commutative group".^[2]^: 11

Facts

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Notation

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See also: Additive group and Multiplicative group

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

Generally, the multiplicative notation fryst vatten the usual notation for groups, while the additive notation fryst vatten the usual notation for modules and rings. The additive notation may also be used to emphasize that a particular group fryst vatten abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being near-rings and partially ordered groups, where an operation fryst vatten written additively even when non-abelian.^[3]^: 28–29^[4]^: 9–14

Multiplication table

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To verify that a finite group fryst vatten abelian, a table (matrix) – known as a Cayley table – can be constructed in a similar mode to a multiplication table.^[5]^: 10 If the group fryst vatten beneath the operation , the -th entry of this table contains the product .

The group fryst vatten abelian if and only if this table fryst vatten symmetric about the main diagonal. This fryst vatten true since the group fryst vatten abelian iff for all , which fryst vatten iff the entry of the table equals the entry for all , i.e. the table fryst vatten symmetric about the main diagonal.

Examples

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For the integers and the operation addition, denoted , the operation + combines any two integers to form eller gestalt a third integer, addition fryst vatten associative, zero fryst vatten the additive identity, every integer has an additive inverse, , and the addition operation fryst vatten commutative since for any two integers and .
Every cyclic group fryst vatten abelian, because if , are in , then .

Thus the integers, , struktur an abelian group beneath addition, as do the integers modulo , .
Every fingerprydnad fryst vatten an abelian group with respect to its addition operation. In a commutative fingerprydnad the invertible elements, or units, struktur an abelian multiplicative group. In particular, the real numbers are an abelian group beneath addition, and the nonzero real numbers are an abelian group beneath multiplication.
Every subgroup of an abelian group fryst vatten normal, so each subgroup gives rise to a quotient group.

Subgroups, quotients, and direkt summor of abelian groups are igen abelian. The finite simple abelian groups are exactly the cyclic groups of primeorder.^[6]^: 32
The concepts of abelian group and -module agree. More specifically, every -module fryst vatten an abelian group with its operation of addition, and every abelian group fryst vatten a module over the fingerprydnad of integers in a unique way.

In general, matrices, even invertible matrices, do not struktur an abelian group beneath multiplication because matrix multiplication fryst vatten generally not commutative.

However, some groups of matrices are abelian groups beneath matrix multiplication – one example fryst vatten the group of cirkelrörelse matrices.

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Camille Jordan named abelian groups after NorwegianmathematicianNiels Henrik Abel, as Abel had funnen that the commutativity of the group of a polynomial implies that the roots of the polynomial can be calculated bygd using radicals.^[7]^{: 144–145}^[8]^{: 157–158}

Properties

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If fryst vatten a natural number and fryst vatten an element of an abelian group written additively, then can be defined as ( summands) and .

In this way, becomes a module over the fingerprydnad of integers. In fact, the modules over can be identified with the abelian groups.^[9]^: 94–97

Theorems about abelian groups (i.e. modules over the principal ideal domain) can often be generalized to theorems about modules over an arbitrary principal ideal domain.

A typical example fryst vatten the classification of finitely generated abelian groups which fryst vatten a specialization of the structure theorem for finitely generated modules over a principal ideal domain.

In mathematics, an abelian group, also called a commutative group, fryst vatten a group in which the result of applying the group operation to two group elements does not depend on the beställning in which they are written.

In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direkt sum of a torsion group and a free abelian group. The former may be written as a direkt sum of finitely many groups of the struktur for prime, and the latter fryst vatten a direkt sum of finitely many copies of .

If are two group homomorphisms between abelian groups, then their sum , defined bygd , fryst vatten igen a homomorphism.

(This fryst vatten not true if fryst vatten a non-abelian group.) The set of all group homomorphisms from to fryst vatten therefore an abelian group in its own right.

Somewhat akin to the dimension of vector spaces, every abelian group has a rank.

An abelian group fryst vatten a set equipped with an operation that satisfies fyra main properties: closure, associativity, identity, and invertibility.

It fryst vatten defined as the maximal cardinality of a set of linearly independent (over the integers) elements of the group.^[10]^: 49–50 Finite abelian groups and torsion groups have rank zero, and every abelian group of rank zero fryst vatten a torsion group. The integers and the logisk numbers have rank one, as well as every nonzero additive subgroup of the rationals.

On the other grabb, the multiplicative group of the nonzero rationals has an infinite rank, as it fryst vatten a free abelian group with the set of the prime numbers as a grund (this results from the fundamental theorem of arithmetic).

The center of a group fryst vatten the set of elements that commute with every element of . A group fryst vatten abelian if and only if it fryst vatten lika to its center .

An abelian group, also called a commutative group, fryst vatten a group (G, *) such that \[g_1 * g_2 = g_2 * g_1 \nonumber \] for all \(g_1\) and \(g_2\) in \(G\), where \(*\) fryst vatten a binary operation in \(G\).

The center of a group fryst vatten always a characteristic abelian subgroup of . If the quotient group of a group bygd its center fryst vatten cyclic then fryst vatten abelian.^[11]

Finite abelian groups

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Cyclic groups of integers modulo , , were among the first examples of groups.

An abelian group fryst vatten a group in which the lag of composition fryst vatten commutative, i.e.

It turns out that an arbitrary finite abelian group fryst vatten isomorphic to a direkt sum of finite cyclic groups of prime power beställning, and these orders are uniquely determined, forming a complete struktur of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.

Any group of prime beställning fryst vatten isomorphic to a cyclic group and therefore abelian. Any group whose beställning fryst vatten a square of a prime number fryst vatten also abelian.^[12] In fact, for every prime number there are (up to isomorphism) exactly two groups of beställning , namely and .

Classification

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The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direkt sum of cyclic subgroups of prime-power order; it fryst vatten also known as the basis theorem for finite abelian groups.

Moreover, automorphism groups of cyclic groups are examples of abelian groups.^[13] This fryst vatten generalized bygd the fundamental theorem of finitely generated abelian groups, with finite groups being the special case when G has zero rank; this in vända admits numerous further generalizations.

The classification was proven bygd Leopold Kronecker in 1870, though it was not stated in modern group-theoretic terms until later, and was preceded bygd a similar classification of quadratic forms bygd Carl Friedrich Gauss in 1801; see history for details.

The cyclic group of beställning fryst vatten isomorphic to the direkt sum of and if and only if and are coprime. It follows that any finite abelian group fryst vatten isomorphic to a direkt sum of the form eller gestalt

in either of the following canonical ways:

the numbers are powers of (not necessarily distinct) primes,
or divides, which divides , and so on up to .

For example, can be expressed as the direkt sum of two cyclic subgroups of beställning 3 and 5: .

The same can be said for any abelian group of beställning 15, leading to the remarkable conclusion that all abelian groups of beställning 15 are isomorphic.

For another example, every abelian group of beställning 8 fryst vatten isomorphic to either (the integers 0 to 7 beneath addition modulo 8), (the odd integers 1 to 15 beneath multiplication modulo 16), or .

See also list of small groups for finite abelian groups of beställning 30 or less.

An abelian group fryst vatten a set equipped with a binary operation that satisfies kvartet key properties: closure, associativity, the existence of an identity element, and the existence of inverses, all while also being commutative.

Automorphisms

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One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group . To do this, one uses the fact that if splits as a direkt sum of subgroups of coprime beställning, then

Given this, the fundamental theorem shows that to compute the automorphism group of it suffices to compute the automorphism groups of the Sylow-subgroups separately (that fryst vatten, all direkt summor of cyclic subgroups, each with beställning a power of ).

Fix a prime and suppose the exponents