Abelian group

Abelian Group (2.2)

In mathematics, a Abelian group or switching group is a group in which the internal operation satisfies the commutative property, that is, that the result of the operation is independent of the order of arguments. More formally, a group ${displaystyle (G,circ)}$ is abelian when, in addition to group axioms, the following condition is satisfied

${displaystyle gcirc h=hcirc g}$for any pair of items ${displaystyle g,hin G}$.

Abelian groups are named after the Norwegian mathematician Niels Henrik Abel, who used these groups in the study of algebraic equations that can be solved by radicals. Groups that are not commutative are called non-abelian or not commutative.

Abelian groups are the basis on which more complex algebraic structures such as rings and fields, vector spaces or modules are built. In category theory, abelian groups are the object of study of the category Ab.

Notation

There are two main notations for abelian groups: additive and multiplicative, described below.

Notation	Operation	Neutral element	Powers	Inverse elements	Direct asthma / Direct output
Addendum	${displaystyle a+b}$	0	${displaystyle na}$	${displaystyle}$	${displaystyle Goplus H}$
Multiplication	${displaystyle a*b}$ or ${displaystyle ab}$	e or 1	${displaystyle a^{n}$	${displaystyle a^{-1}$or${displaystyle {frac {1}{a}}}}$	${displaystyle Gtimes H}$

The multiplicative notation is the usual notation in group theory, while the additive is the usual notation in the study of rings, modules and vector spaces, in which there is a second operation. It is also common to use the additive notation when working only with abelian groups, as in the case of homological algebra.

Examples

All cyclical groups ${displaystyle G=langle arangle }$ is abelian, for two elements whatsoever ${displaystyle x,yin G}$ can be expressed as powers ${displaystyle x=a^{m}, y=a^{n}}$ for certain integers m and n. Consequently

${displaystyle xy=a^{m} a^{n}=a^{m+n}=a^{n+}=a^{n} a^{m}=yx}$.

In particular, the additive group of integers ${displaystyle mathbb {Z} }$ is abelian, like the whole group module n, ${displaystyle mathbb {Z} _{n}}$.

The rational numbers, the real numbers, the complex numbers, and the quaternions are each an abelian group under addition. They are also under multiplication (excluding zero from each of these sets) except for quaternions, which are a notable example of a non-commutative field. In general, every ring is an abelian group with respect to addition. If addition is a commutative ring, the invertible elements also form an abelian group under multiplication.

Given a group ${displaystyle G}$ arbitrary, it is possible to build abelianization of ${displaystyle G}$, which is the quotient of ${displaystyle G}$ by its switching subgroup: ${displaystyle G/[G,G]}$. This group is abelian, and it has the property that if given any other normal subgroup ${displaystyle N}$, the quotient ${displaystyle G/N}$ He's abelian, then ${displaystyle [G,G]subset N}$.

All group ${displaystyle G}$ contains an abelian subgroup called the center of the group, which is formed by the elements that commut with any other group.

Properties

• Yeah. n is a natural number and x an element of an Abelian group G (with additive notation), you can define nx = x + x +... + x (n and (−n)x = −(nx) with what G becomes a module on the ring Z of the whole. In fact, modules on Z It's not other than the Abelian groups.
• Yeah. f, g: G → H are two homomorphisms between Abelian groups, their sum (defined by (f + g)(x) = f(x+ g(x) is also a homomorphism; this is not generally fulfilled for non-Abellian groups. With this operation, the set of homomorphisms between G and H becomes, then, an Abelian group in itself.
• Every subgroup of an Abelian group is normal, and therefore for every subgroup there is a quotient group. Subgroups, quotient groups, and direct sums of Abelian groups are also Abelians.

Classification of finitely generated abelian groups

A group is said to be finitely generated if there exists a spanning set of the group that is finite. Every finite group is finitely generated, since the group itself is a generating set of itself. Finite and finitely generated abelian groups are fully classified by the so-called structure theorem, of which there are several versions. According to this theorem, every finitely generated abelian group is the direct sum of cyclic groups, which can be of two types:

• the infinite cyclic group, characterized by the integers under the addition, ${displaystyle mathbb {Z} }$.
• finite cyclic groups, characterized by the integer module n under the sum module n, ${displaystyle mathbb {Z} _{n}}$.

Finite abelian groups

It is interesting to study first the case of finite groups, since this result applies directly to the general case. The structure theorem in the finite case states the following:

Numbers ${displaystyle d_{1},ldotsd_{t}}$ called torsion coefficients of ${displaystyle G}$And they're invariants of the group. In particular, the order ${displaystyle G}$ equals the product ${displaystyle d_{1}ldots d_{t}}$. It is said that an element ${displaystyle g}$ of a group is an element of torsion if your order is finite. Similarly, it is said that a group in which all the elements are of torsion is a torsion group. Naturally, all finite groups are torsion.

This theorem is deducted from the following result, using which ${displaystyle mathbb {Z} _{m}oplus mathbb {Z} _{n}}}$ isomorph to ${displaystyle mathbb {Z} _{nm}}$ When n and m They're coprimos:

The following examples illustrate how to apply the structure theorem, from the prime factors of the order of the group:

• Except isomorphism, there are five Abelian groups with 16 elements. From which ${displaystyle 16=2^{4}}$, possible choices for torsion coefficients are ${displaystyle (16),(2times 8),(2times 2times 4),(2times 2times 2times 2times 2){mbox{ and }}(4times 4)}$. Consequently, an Abelian group of 16 elements is isomorphous to one and only one of the following:

${displaystyle mathbb {Z} _{16},quad mathbb {Z} _{2}oplus mathbb {Z}{8},quad mathbb {Z}{2}{2}{obb}}{2}{2}{2}{2}{x1⁄2}}{obb}}{oblus}$.

• Every Abelian group of order 30 is isomorph to the cyclic group ${displaystyle mathbb {Z} _{30}simeq mathbb {Z} _{5}oplus mathbb {Z} _{3}oplus mathbb {Z} _{2}}}$. This is because there is no way to break down 30 as a product of two larger numbers of 1 such that one is dividing the other.

Finitely generated abelian groups

The set of torsion elements of an arbitrary group form a subgroup called subgroup of torsionand denotes as ${displaystyle tau (G)}$. If the only element of torsion is identity, then the group is said to be free torsion. In such a case, any element other than identity is of infinite order. The following result indicates how an Abelian group can be broken down into two parts: a torsion and a torsion free:

If the Abelian group ${displaystyle G}$ is finitely generated then its torsion subgroup is also finitely generated, and in fact it is finite. It can therefore be classified according to the preceding section. Plus ${displaystyle G=tau (G)oplus F}$Where ${displaystyle F}$ is a finitely generated and torsion-free abelian group. The following result allows us to characterize this group ${displaystyle F}$:

A Finnishly Generated Abelian Group ${displaystyle G}$ is a free abelian group if it is isomorphic to the direct product ${displaystyle mathbb {Z} ^{n}$for a certain positive integer ${displaystyle n}$, called range of ${displaystyle G}$. Consequently

In summary, every finitely generated abelian group is isomorphic to the direct sum

${displaystyle mathbb {Z} oplus...oplus mathbb {Z} oplus mathbb {Z} _{d_{1}}oplus...oplus mathbb {Z} _{d_{t}}}}}$

where the number of factors ${displaystyle mathbb {Z} }$ is rank and numbers ${displaystyle d_{1},ldotsd_{t}}$ are the torsion coefficients of ${displaystyle G}$to verify that ${displaystyle d_{i}cultd_{i+1}, forall i=1,ldotst-1}$.