Abelian group

An abelian (or commutative ) group is a group in which a group operation is commutative ; in other words, the group $(G,\;*)$ ${\ displaystyle (G, \; *)}$ ${\ displaystyle (G, \; *)}$ abelian if $a*b=b*a$ ${\ displaystyle a * b = b * a}$ $a * b = b * a$ for any two elements $a,\;b\in G$ ${\ displaystyle a, \; b \ in G}$ $a, \; b \ in G$ .

Typically, an abelian group uses an additive notation to denote a group operation, i.e., a group operation is indicated by $+$ ${\ displaystyle +}$ $+$ and is called addition ^[1] .

The name is given in honor of the Norwegian mathematician Niels Abel .

Content

1 Examples
2 Related Definitions
3 Properties
4 Finite Abelian groups
5 Variations and generalizations
6 See also
7 notes
8 Literature

Examples

A group of parallel translations in linear space.
Any cyclic group $G$ $=$ $⟨$ $a$ $⟩$ {\ displaystyle G = \ langle a \ rangle} abelian. Indeed, for any $x$ $=$ $a$ $n$ {\ displaystyle x = a ^ {n}} and $y$ $=$ $a$ $m$ {\ displaystyle y = a ^ {m}} it is true that
$xy=a^{m}a^{n}=a^{m+n}=a^{n}a^{m}=yx$ ${\ displaystyle xy = a ^ {m} a ^ {n} = a ^ {m + n} = a ^ {n} a ^ {m} = yx}$ .
- In particular, many $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ integers are a commutative addition group; the same is true for residue classes $\mathbb {Z} /n\mathbb {Z} \,.$ ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z} \ ,.}$
Any ring is a commutative (abelian) group in its composition; an example is the field $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ real numbers with the operation of adding numbers.
Invertible elements of a commutative ring (in particular, nonzero elements of any field ) form an Abelian group by multiplication. For example, an abelian group is the set of nonzero real numbers with the operation of multiplication.

Related Definitions

By analogy with the dimension of vector spaces , each abelian group has a rank . It is defined as the minimum dimension of a vector space over a field $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ rational numbers into which the torsion factor of the group is embedded.

Properties

Finitely generated abelian groups are isomorphic to direct sums of cyclic groups .
- Finite Abelian groups are isomorphic to direct sums of finite cyclic groups.
Any Abelian group has a natural module structure over a ring of integers . Indeed let $n$ {\ displaystyle n} Is a natural number , and $x$ {\ displaystyle x} Is an element of the commutative group $G$ {\ displaystyle G} with the operation denoted by +, then $n$ $x$ {\ displaystyle nx} can be defined as $x$ $+$ $x$ $+$ $...$ $+$ $x$ {\ displaystyle x + x + \ ldots + x} ( $n$ {\ displaystyle n} times) and $($ $-$ $n$ $)$ $x$ $=$ $-$ $($ $n$ $x$ $)$ {\ displaystyle (-n) x = - (nx)} .
- Statements and theorems that are true for Abelian groups (i.e., modules over the domain of principal ideals $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ ), can often be generalized to modules over an arbitrary domain of principal ideals. A typical example is the classification of finitely generated abelian groups , which can be generalized to the classification of arbitrary finitely generated modules over a domain of principal ideals .
Many homomorphisms $\operatorname {Hom} (G,\;H)$ ${\ displaystyle \ operatorname {Hom} (G, \; H)}$ all group homomorphisms from $G$ ${\ displaystyle G}$ at $H$ ${\ displaystyle H}$ it is itself an abelian group. Indeed let $f,\;g:G\to H$ ${\ displaystyle f, \; g: G \ to H}$ Are two homomorphisms of groups between abelian groups, then their sum $f+g$ ${\ displaystyle f + g}$ set as $(f+g)(x)=f(x)+g(x)$ ${\ displaystyle (f + g) (x) = f (x) + g (x)}$ , is also a homomorphism (this is not true if $H$ ${\ displaystyle H}$ is not a commutative group).
The concept of Abelianity is closely related to the concept of a center $Z(G)$ ${\ displaystyle Z (G)}$ groups $G$ ${\ displaystyle G}$ - a set consisting of those elements of it that commute with each element of the group $G$ ${\ displaystyle G}$ , and playing the role of a kind of "measure of Abelianity." A group is Abelian if and only if its center coincides with the whole group.

Finite Abelian groups

The fundamental theorem on the structure of a finite abelian group claims that any finite abelian group can be decomposed into the direct sum of its cyclic subgroups whose orders are powers of primes . This is a consequence of the general theorem on the structure of finitely generated abelian groups for the case when the group does not have elements of infinite order. $\mathbb {Z} _{mn}$ ${\ displaystyle \ mathbb {Z} _ {mn}}$ isomorphic to a direct sum $\mathbb {Z} _{m}$ ${\ displaystyle \ mathbb {Z} _ {m}}$ and $\mathbb {Z} _{n}$ ${\ displaystyle \ mathbb {Z} _ {n}}$ if and only if $m$ ${\ displaystyle m}$ and $n$ ${\ displaystyle n}$ mutually simple.

Therefore, we can write an Abelian group $G$ ${\ displaystyle G}$ in the form of a direct amount

\mathbb {Z} _{k_{1}}\oplus \ldots \oplus \mathbb {Z} _{k_{u}}

{\ displaystyle \ mathbb {Z} _ {k_ {1}} \ oplus \ ldots \ oplus \ mathbb {Z} _ {k_ {u}}}

in two different ways:

Where are the numbers $k_{1},\;\ldots ,\;k_{u}$ ${\ displaystyle k_ {1}, \; \ ldots, \; k_ {u}}$ degrees of simple
Where $k_{1}$ ${\ displaystyle k_ {1}}$ divides $k_{2}$ ${\ displaystyle k_ {2}}$ which divides $k_{3}$ ${\ displaystyle k_ {3}}$ and so on until $k_{u}$ ${\ displaystyle k_ {u}}$ .

For example, $\mathbb {Z} /15\mathbb {Z} =\mathbb {Z} _{15}$ ${\ displaystyle \ mathbb {Z} / 15 \ mathbb {Z} = \ mathbb {Z} _ {15}}$ can be decomposed into the direct sum of two cyclic subgroups of orders 3 and 5: $\mathbb {Z} /15\mathbb {Z} =\{0,\;5,\;10\}\oplus \{0,\;3,\;6,\;9,\;12\}$ ${\ displaystyle \ mathbb {Z} / 15 \ mathbb {Z} = \ {0, \; 5, \; 10 \} \ oplus \ {0, \; 3, \; 6, \; 9, \; 12 \}}$ . The same can be said of any Abelian group of the order of fifteen; as a result, we conclude that all abelian groups of order 15 are isomorphic.

Variations and generalizations

A differential group is an abelian group. $\mathbf {C}$ ${\ displaystyle \ mathbf {C}}$ in which such an endomorphism is given $d\colon \mathbf {C} \to \mathbf {C}$ ${\ displaystyle d \ colon \ mathbf {C} \ to \ mathbf {C}}$ , what $d^{2}=0$ ${\ displaystyle d ^ {2} = 0}$ . This endomorphism is called a differential . Elements of differential groups are called chains , elements of the kernel $\ker \,d$ ${\ displaystyle \ ker \, d}$ - cycles , image elements $\mathrm {Im} \,d$ ${\ displaystyle \ mathrm {Im} \, d}$ - by the borders .
A ring is an Abelian group on which an additional binary operation of “multiplication” is given, satisfying distributive axioms .
A metabelian group is a group whose commutant is abelian.
A nilpotent group is a group whose central series is finite.
A solvable group is a group whose series of commutants stabilizes on a trivial group.
A Dedekind group is a group whose every subgroup is normal .

Notes

↑ Abelian group - an article from the Mathematical Encyclopedia . Yu. L. Ershov

Literature

Vinberg E. B. The course of algebra. - 3rd ed. - M .: Factorial Press, 2002 .-- 544 p. - 3000 copies. - ISBN 5-88688-060-7 . .
Fuchs L. Infinite Abelian groups. - World, 1974.

[1] Abelian group - an article from the Mathematical Encyclopedia . Yu. L. Ershov