Difference between revisions of "Abelian group"

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This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a group property that is pivotal (i.e., important) among existing group properties
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RANDOM GROUP PROPERTY: Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.

History

Origin of the term

The term abelian group comes from Niels Henrick Abel, a mathematician who worked with groups even before the formal theory was laid down, in order to prove unsolvability of the quintic.

The word abelian is usually begun with a small a.

wikinote: Some older content on the wiki uses capital A for Abelian. We're trying to update this content.

Definition

An abelian group is a group where any two elements commute. In symbols, a group $G$ is termed abelian if for any elements $x$ and $y$ in $G$ , $xy = yx$ (here $xy$ denotes the product of $x$ and $y$ in $G$ ). Note that $x,y$ are allowed to be equal, though equal elements commute anyway, so we can restrict attention if we wish to unequal elements.

Full definition

An abelian group is a set $G$ equipped with a (infix) binary operation $+$ (called the addition or group operation), an identity element $0$ and a (prefix) unary operation $-$ , called the inverse map or negation map, satisfying the following:

For any $a,b,c \in G$ , $a + (b + c) = (a + b) + c$ . This property is termed associativity.
For any $a \in G$ , $a + 0 = 0 + a = a$ . $0$ thus plays the role of an additive identity element or neutral element.
For any $a \in G$ , $a + (-a) = (-a) + a = 0$ . Thus, $-a$ is an inverse element to $a$ with respect to $+$ .
For any $a,b \in G$ , $a + b = b + a$ . This property is termed commutativity.

Equivalent formulations

A group $G$ is termed abelian if it satisfies the following equivalent conditions:

Its center $Z(G)$ is the whole group.
Its derived subgroup $G' = [G,G]$ is trivial.
(Choose a generating set $S$ for $G$ ). For any elements $a,b \in S$ , $ab = ba$ .

Notation

When $G$ is an abelian group, we typically use additive notation and terminology. Thus, the group multiplication is termed addition and the product of two elements is termed the sum.

The infix operator $+$ is used for the group multiplication, so the sum of two elements $a$ and $b$ is denoted by $a + b$ . The group multiplication is termed addition and the product of two elements is termed the sum.
The identity element is typically denoted as $0$ and termed zero
The inverse of an element is termed its negative or additive inverse. The inverse of $a$ is denoted $-a$
$a + a + \ldots + a$ done $n$ times is denoted $na$ , (where $n \in \mathbb{N}$ ) while $(-a) + (-a) + (-a) + \ldots + (-a)$ done $n$ times is denoted $(-n)a$ .

This convention is typically followed in a situation where we are dealing with the abelian group $G$ in isolation, rather than as a subgroup of a possibly non-abelian group. If we are working with subgroups in a non-abelian group, we typically use multiplicative notation even if the subgroup happens to be abelian.

Examples

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Some infinite examples

The additive group of integers $\mathbb{Z}$ , the additive group of rational numbers $\mathbb{Q}$ , the additive group of real numbers $\mathbb{R}$ , the multiplicative group of nonzero rationals $\mathbb{Q}^*$ , and the multiplicative group of nonzero real numbers $\mathbb{R}^*$ are some examples of Abelian groups.

(More generally, for any field, the additive group, and the multiplicative group of nonzero elements, are Abelian groups).

Finite examples

Cyclic groups are good examples of abelian groups, where the cyclic group of order $n$ is the group of integers modulo $n$ .

Further, any direct product of cyclic groups is also an abelian group. Further, every finitely generated Abelian group is obtained this way. This is the famous structure theorem for finitely generated abelian groups.

The structure theorem can be used to generate a complete listing of finite abelian groups, as described here: classification of finite Abelian groups.

Non-examples

Not every group is abelian. The smallest non-abelian group is the symmetric group on three letters: the group of all permutations on three letters, under composition. Its being non-abelian hinges on the fact that the order in which permutations are performed matters.

Facts

Occurrence as subgroups

Every cyclic group is abelian. Since each group is generated by its cyclic subgroups, every group is generated by a family of abelian subgroups. A trickier question is: do there exist abelian normal subgroups? A good candidate for an abelian normal subgroup is the center, which is the collection of elements of the group that commute with every element of the group.

Occurrence as quotients

The maximal abelian quotient of any group is termed its abelianization, and this is the quotient by the derived subgroup. A subgroup is an abelian-quotient subgroup (i.e., normal with abelian quotient group) if and only if the subgroup contains the derived subgroup.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
varietal group property	Yes		The collection of abelian groups forms a subvariety of the variety of groups. In particular, it is closed under taking subgroups, quotients, and arbitrary direct products
subgroup-closed group property	Yes	abelianness is subgroup-closed	If $G$ is an abelian group and $H$ is a subgroup of $G$ , then $H$ is abelian.
quotient-closed group property	Yes	abelianness is quotient-closed	If $G$ is an abelian group and $H$ is a normal subgroup of $G$ , the quotient group $G/H$ is abelian.
direct product-closed group property	Yes	abelianness is direct product-closed	Suppose $G_i, i \in I$ , are abelian groups. Then, the external direct product $\prod_{i \in I} G_i$ is also abelian.

Formalisms

In terms of the diagonal-in-square operator

This property is obtained by applying the diagonal-in-square operator to the property: normal subgroup
View other properties obtained by applying the diagonal-in-square operator

A group $G$ is an abelian group if and only if, in the external direct product $G \times G$ , the diagonal subgroup $\{ (g,g) \mid g \in G \}$ is a normal subgroup.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
cyclic group	generated by one element	cyclic implies abelian	abelian not implies cyclic (see also list of examples)	Epabelian group, Locally cyclic group, Residually cyclic group\|FULL LIST, MORE INFO
homocyclic group	direct product of isomorphic cyclic groups		(see also list of examples)	\|FULL LIST, MORE INFO
residually cyclic group	every non-identity element is outside a normal subgroup with a cyclic quotient group		(see also list of examples)	\|FULL LIST, MORE INFO
locally cyclic group	every finitely generated subgroup is cyclic		(see also list of examples)	Epabelian group\|FULL LIST, MORE INFO
epabelian group	abelian group whose exterior square is the trivial group		(see also list of examples)	\|FULL LIST, MORE INFO
finite abelian group	abelian and a finite group		(see also list of examples)	\|FULL LIST, MORE INFO
finitely generated abelian group	abelian and a finitely generated group		(see also list of examples)	Residually cyclic group\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nilpotent group	lower central series reaches identity, upper central series reaches whole group	abelian implies nilpotent	nilpotent not implies abelian (see also list of examples)	Group in which class equals maximum subnormal depth, Group of nilpotency class three, Group of nilpotency class two, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, UL-equivalent group\|FULL LIST, MORE INFO
solvable group	derived series reaches identity, has normal series with abelian factor groups	abelian implies solvable	solvable not implies abelian (see also list of examples)	Metabelian group, Metanilpotent group, Nilpotent group\|FULL LIST, MORE INFO
metabelian group	has abelian normal subgroup with abelian quotient group		(see also list of examples)	Group of nilpotency class two\|FULL LIST, MORE INFO
virtually abelian group	has abelian subgroup of finite index		(see also list of examples)	FZ-group\|FULL LIST, MORE INFO

Incomparable properties

Supersolvable group is a group that has a normal series where all the successive quotient groups are cyclic groups. An abelian group is supersolvable if and only if it is finitely generated.
Polycyclic group is a group that has a subnormal series where all the successive quotent groups are cyclic groups. An abelian group is polycyclic if and only if it is finitely generated.

Testing

The testing problem

Further information: Abelianness testing problem

The abelianness testing problem is the problem of testing whether a group (described using some group description rule, such as an encoding of a group or a multi-encoding of a group) is abelian.

Algorithms for the abelianness testing problem range from the brute-force black-box group algorithm for abelianness testing (that involves testing for every pair of elements whether they commute, and is quadratic in the order of the group) to the generating set-based black-box group algorithm for abelianness testing (that involves testing only on a generating set, and is quadratic in the size of the generating set).

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsAbelian
The class of all groups with this property can be referred to with the built-in command: AbelianGroups
View GAP-testable group properties

To test whether a group is abelian, the GAP syntax is:

IsAbelian (group)

where group either defines the group or gives the name to a group previously defined.

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20K

References

Textbook references

Book

Page number

Chapter and section

Contextual information

View

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info}

17

Formal definition (definition as point (2) in general definition of group)

Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info}

2

1.1 (Rudiments of Group Theory/Review)

definition introduced in paragraph

Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632^{More info}

42

definition introduced in paragraph (immediately after definition of group)

Topics in Algebra by I. N. Herstein^{More info} || 28 || || Formal definition
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info} || 2 || 1.1 (Binary Operations, Semigroups, and Groups) || formal definition
Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, Page 1, 1.1 (Elementary group theory), (definition introduced in paragraph)^{More info}

External links

Definition links

Perspective links

Chapman page (from the algebraic structures viewpoint)

@@ Line 82: / Line 82: @@
 ===Occurrence as quotients===
-The maximal abelian quotient of any group is termed its [[abelianization]], and this is the quotient by the [[derived subgroup]]. A subgroup is an [[abelian-quotient subgroup]] (i.e., normal with abelian quotient group) if and only if the subgroup contains the commutator subgroup.
+The maximal abelian quotient of any group is termed its [[abelianization]], and this is the quotient by the [[derived subgroup]]. A subgroup is an [[abelian-quotient subgroup]] (i.e., normal with abelian quotient group) if and only if the subgroup contains the derived subgroup.
 ==Metaproperties==