Glossary of group theory

Last updated July 30, 2024

A group is a set together with an associative operation that admits an identity element and such that there exists an inverse for every element.

A

abelian group: A group $(G, •)$ is abelian if $•$ is commutative, i.e. $g • h = h • g$ for all $g, h \in G$ . Likewise, a group is nonabelian if this relation fails to hold for any pair $g, h \in G$ .
ascendant subgroup: A subgroup $H$ of a group $G$ is ascendant if there is an ascending subgroup series starting from $H$ and ending at $G$ , such that every term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup is subnormal.
automorphism: An automorphism of a group is an isomorphism of the group to itself.

C

center of a group

The center of a group

G

, denoted

Z(G)

, is the set of those group elements that commute with all elements of

G

, that is, the set of all

h \in G

such that

hg = gh

for all

g \in G

.

Z(G)

is always a normal subgroup of

G

. A group

G

is abelian if and only if

Z(G) = G

.

centerless group

A group

G

is centerless if its center

Z(G)

is trivial.

central subgroup

A subgroup of a group is a central subgroup of that group if it lies inside the center of the group.

centralizer

For a subset

S

of a group

G

, the centralizer of

S

in

G

, denoted

C G (S)

, is the subgroup of

G

defined by

\mathrm {C} _{G}(S)=\{g\in G\mid gs=sg{\text{ for all }}s\in S\}.

characteristic subgroup

A subgroup of a group is a characteristic subgroup of that group if it is mapped to itself by every automorphism of the parent group.

characteristically simple group

A group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups.

class function

A class function on a group

G

is a function that it is constant on the conjugacy classes of

G

.

class number

The class number of a group is the number of its conjugacy classes.

commutator

The commutator of two elements

g

and

h

of a group

G

is the element

[g, h] = g -1 h -1 gh

. Some authors define the commutator as

[g, h] = ghg -1 h -1

instead. The commutator of two elements

g

and

h

is equal to the group's identity if and only if

g

and

h

commutate, that is, if and only if

gh = hg

.

commutator subgroup

The commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

composition series

A composition series of a group

G

is a subnormal series of finite length

1=H_{0}\triangleleft H_{1}\triangleleft \cdots \triangleleft H_{n}=G,

with strict inclusions, such that each

H i

is a maximal strict normal subgroup of

H i +1

. Equivalently, a composition series is a subnormal series such that each factor group

H i +1 / H i

is simple. The factor groups are called composition factors.

conjugacy-closed subgroup

A subgroup of a group is said to be conjugacy-closed if any two elements of the subgroup that are conjugate in the group are also conjugate in the subgroup.

conjugacy class

The conjugacy classes of a group

G

are those subsets of

G

containing group elements that are conjugate with each other.

conjugate elements

Two elements

x

and

y

of a group

G

are conjugate if there exists an element

g \in G

such that

g -1 xg = y

. The element

g -1 xg

, denoted

x g

, is called the conjugate of

x

by

g

. Some authors define the conjugate of

x

by

g

as

gxg -1

. This is often denoted

g x

. Conjugacy is an equivalence relation. Its equivalence classes are called conjugacy classes.

conjugate subgroups

Two subgroups

H 1

and

H 2

of a group

G

are conjugate subgroups if there is a

g \in G

such that

gH 1 g -1 = H 2

.

contranormal subgroup

A subgroup of a group

G

is a contranormal subgroup of

G

if its normal closure is

G

itself.

cyclic group

A cyclic group is a group that is generated by a single element, that is, a group such that there is an element

g

in the group such that every other element of the group may be obtained by repeatedly applying the group operation to

g

or its inverse.

D

derived subgroup: Synonym for commutator subgroup.
direct product: The direct product of two groups $G$ and $H$ , denoted $G \times H$ , is the cartesian product of the underlying sets of $G$ and $H$ , equipped with a component-wise defined binary operation $(g 1, h 1) \cdot (g 2, h 2) = (g 1 \cdot g 2, h 1 \cdot h 2)$ . With this operation, $G \times H$ itself forms a group.

E

exponent of a group: The exponent of a group $G$ is the smallest positive integer $n$ such that $g n = e$ for all $g \in G$ . It is the least common multiple of the orders of all elements in the group. If no such positive integer exists, the exponent of the group is said to be infinite.

F

factor group: Synonym for quotient group.
FC-group: A group is an FC-group if every conjugacy class of its elements has finite cardinality.
finite group: A finite group is a group of finite order, that is, a group with a finite number of elements.
finitely generated group: A group $G$ is finitely generated if there is a finite generating set, that is, if there is a finite set $S$ of elements of $G$ such that every element of $G$ can be written as the combination of finitely many elements of $S$ and of inverses of elements of $S$ .

G

generating set: A generating set of a group $G$ is a subset $S$ of $G$ such that every element of $G$ can be expressed as a combination (under the group operation) of finitely many elements of $S$ and inverses of elements of $S$ . Given a subset $S$ of $G$ . We denote by $⟨ S ⟩$ the smallest subgroup of $G$ containing $S$ . $⟨ S ⟩$ is called the subgroup of $G$ generated by $S$ .
group automorphism: See automorphism.
group homomorphism: See homomorphism.
group isomorphism: See isomorphism.

H

homomorphism: Given two groups $(G, •)$ and $(H, \cdot)$ , a homomorphism from $G$ to $H$ is a function $h : G \to H$ such that for all $a$ and $b$ in $G$ , $h (a • b) = h (a) \cdot h (b)$ .

I

index of a subgroup: The index of a subgroup $H$ of a group $G$ , denoted $| G : H |$ or $[G : H]$ or $(G : H)$ , is the number of cosets of $H$ in $G$ . For a normal subgroup $N$ of a group $G$ , the index of $N$ in $G$ is equal to the order of the quotient group $G / N$ . For a finite subgroup $H$ of a finite group $G$ , the index of $H$ in $G$ is equal to the quotient of the orders of $G$ and $H$ .
isomorphism: Given two groups $(G, •)$ and $(H, \cdot)$ , an isomorphism between $G$ and $H$ is a bijective homomorphism from $G$ to $H$ , that is, a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. Two groups are isomorphic if there exists a group isomorphism mapping from one to the other. Isomorphic groups can be thought of as essentially the same, only with different labels on the individual elements.

L

lattice of subgroups: The lattice of subgroups of a group is the lattice defined by its subgroups, partially ordered by set inclusion.
locally cyclic group: A group is locally cyclic if every finitely generated subgroup is cyclic. Every cyclic group is locally cyclic, and every finitely-generated locally cyclic group is cyclic. Every locally cyclic group is abelian. Every subgroup, every quotient group and every homomorphic image of a locally cyclic group is locally cyclic.

N

no small subgroup

A topological group has no small subgroup if there exists a neighborhood of the identity element that does not contain any nontrivial subgroup.

normal closure

The normal closure of a subset

S

of a group

G

is the intersection of all normal subgroups of

G

that contain

S

.

normal core

The normal core of a subgroup

H

of a group

G

is the largest normal subgroup of

G

that is contained in

H

.

normal series

A normal series of a group

G

is a sequence of normal subgroups of

G

such that each element of the sequence is a normal subgroup of the next element:

1=A_{0}\triangleleft A_{1}\triangleleft \cdots \triangleleft A_{n}=G

with

A_{i}\triangleleft G

.

normal subgroup

A subgroup

N

of a group

G

is normal in

G

(denoted

N ◅ G

) if the conjugation of an element

n

of

N

by an element

g

of

G

is always in

N

, that is, if for all

g \in G

and

n \in N

,

gng -1 \in N

. A normal subgroup

N

of a group

G

can be used to construct the quotient group

G / N

.

normalizer

For a subset

S

of a group

G

, the normalizer of

S

in

G

, denoted

N G (S)

, is the subgroup of

G

defined by

\mathrm {N} _{G}(S)=\{g\in G\mid gS=Sg\}.

O

orbit: Consider a group $G$ acting on a set $X$ . The orbit of an element $x$ in $X$ is the set of elements in $X$ to which $x$ can be moved by the elements of $G$ . The orbit of $x$ is denoted by $G \cdot x$
order of a group: The order of a group $(G, •)$ is the cardinality (i.e. number of elements) of $G$ . A group with finite order is called a finite group.
order of a group element: The order of an element $g$ of a group $G$ is the smallest positive integer $n$ such that $g n = e$ . If no such integer exists, then the order of $g$ is said to be infinite. The order of a finite group is divisible by the order of every element.

P

perfect core: The perfect core of a group is its largest perfect subgroup.
perfect group: A perfect group is a group that is equal to its own commutator subgroup.
periodic group: A group is periodic if every group element has finite order. Every finite group is periodic.
permutation group: A permutation group is a group whose elements are permutations of a given set $M$ (the bijective functions from set $M$ to itself) and whose group operation is the composition of those permutations. The group consisting of all permutations of a set $M$ is the symmetric group of $M$ .
p-group: If $p$ is a prime number, then a $p$ -group is one in which the order of every element is a power of $p$ . A finite group is a $p$ -group if and only if the order of the group is a power of $p$ .
p-subgroup: A subgroup that is also a $p$ -group. The study of $p$ -subgroups is the central object of the Sylow theorems.

Q

quotient group: Given a group $G$ and a normal subgroup $N$ of $G$ , the quotient group is the set $G / N$ of left cosets ${aN : a \in G}$ together with the operation $aN • bN = abN$ . The relationship between normal subgroups, homomorphisms, and factor groups is summed up in the fundamental theorem on homomorphisms.

R

real element: An element $g$ of a group $G$ is called a real element of $G$ if it belongs to the same conjugacy class as its inverse, that is, if there is a $h$ in $G$ with $g h = g -1$ , where $g h$ is defined as $h -1 gh$ . An element of a group $G$ is real if and only if for all representations of $G$ the trace of the corresponding matrix is a real number.

S

serial subgroup

A subgroup

H

of a group

G

is a serial subgroup of

G

if there is a chain

C

of subgroups of

G

from

H

to

G

such that for each pair of consecutive subgroups

X

and

Y

in

C

,

X

is a normal subgroup of

Y

. If the chain is finite, then

H

is a subnormal subgroup of

G

.

simple group

A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

subgroup

A subgroup of a group

G

is a subset

H

of the elements of

G

that itself forms a group when equipped with the restriction of the group operation of

G

to

H \times H

. A subset

H

of a group

G

is a subgroup of

G

if and only if it is nonempty and closed under products and inverses, that is, if and only if for every

a

and

b

in

H

,

ab

and

a -1

are also in

H

.

subgroup series

A subgroup series of a group

G

is a sequence of subgroups of

G

such that each element in the series is a subgroup of the next element:

1=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G.

subnormal subgroup

A subgroup

H

of a group

G

is a subnormal subgroup of

G

if there is a finite chain of subgroups of the group, each one normal in the next, beginning at

H

and ending at

G

.

symmetric group

Given a set

M

, the symmetric group of

M

is the set of all permutations of

M

(the set all bijective functions from

M

to

M

) with the composition of the permutations as group operation. The symmetric group of a finite set of size

n

is denoted

S n

. (The symmetric groups of any two sets of the same size are isomorphic.)

T

torsion group: Synonym for periodic group.
transitively normal subgroup: A subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group.
trivial group: A trivial group is a group consisting of a single element, namely the identity element of the group. All such groups are isomorphic, and one often speaks of the trivial group.

Basic definitions

Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem.

Kernel of a group homomorphism. It is the preimage of the identity in the codomain of a group homomorphism. Every normal subgroup is the kernel of a group homomorphism and vice versa.

Direct product , direct sum , and semidirect product of groups. These are ways of combining groups to construct new groups; please refer to the corresponding links for explanation.

Types of groups

Finitely generated group . If there exists a finite set $S$ such that $⟨ S ⟩ = G$ , then $G$ is said to be finitely generated. If $S$ can be taken to have just one element, $G$ is a cyclic group of finite order, an infinite cyclic group, or possibly a group ${e}$ with just one element.

Simple group . Simple groups are those groups having only $e$ and themselves as normal subgroups. The name is misleading because a simple group can in fact be very complex. An example is the monster group, whose order is about 10⁵⁴. Every finite group is built up from simple groups via group extensions, so the study of finite simple groups is central to the study of all finite groups. The finite simple groups are known and classified.

The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups. This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set.

The situation is much more complicated for the non-abelian groups.

Free group . Given any set $A$ , one can define a group as the smallest group containing the free semigroup of $A$ . The group consists of the finite strings (words) that can be composed by elements from $A$ , together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance $(abb) • (bca) = abbbca$ .

Every group $(G, •)$ is basically a factor group of a free group generated by $G$ . Refer to Presentation of a group for more explanation. One can then ask algorithmic questions about these presentations, such as:

Do these two presentations specify isomorphic groups?; or
Does this presentation specify the trivial group?

The general case of this is the word problem, and several of these questions are in fact unsolvable by any general algorithm.

General linear group , denoted by $GL(n, F)$ , is the group of $n$ -by- $n$ invertible matrices, where the elements of the matrices are taken from a field $F$ such as the real numbers or the complex numbers.

Group representation (not to be confused with the presentation of a group). A group representation is a homomorphism from a group to a general linear group. One basically tries to "represent" a given abstract group as a concrete group of invertible matrices, which is much easier to study.

Related Research Articles

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup $of the group is normal in if and only if for all and . The usual notation for this relation is .$

In group theory, a branch of mathematics, given a group $G$ under a binary operation ∗, a subset $H$ of $G$ is called a subgroup of $G$ if $H$ also forms a group under the operation ∗. More precisely, $H$ is a subgroup of $G$ if the restriction of ∗ to $H \times H$ is a group operation on $H$ . This is often denoted $H \leq G$ , read as " $H$ is a subgroup of $G$ ".

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group $defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .$

<span class="mw-page-title-main">Conjugacy class</span> In group theory, equivalence class under the relation of conjugation

In mathematics, especially group theory, two elements $and of a group are conjugate if there is an element in the group such that This is an equivalence relation whose equivalence classes are called conjugacy classes . In other words, each conjugacy class is closed under for all elements in the group.$

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of $n$ elements is called the alternating group of degree $n$ , or the alternating group on $n$ letters and denoted by $A n$ or $Alt(n).$

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.

In mathematics, the free groupF_S over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms. The members of S are called generators of F_S, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to F_S for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses.

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G. The index is denoted $or or . Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula$

In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.

In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element $a$ of a group, is thus the smallest positive integer $m$ such that $a m = e$ , where $e$ denotes the identity element of the group, and $a m$ denotes the product of $m$ copies of $a$ . If no such $m$ exists, the order of $a$ is infinite.

In mathematics, specifically group theory, a subgroup series of a group $is a chain of subgroups:$

In mathematics, a quasisimple group is a group that is a perfect central extension E of a simple group S. In other words, there is a short exact sequence

In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius.

In mathematics, specifically in group theory, the direct product is an operation that takes two groups $G$ and $H$ and constructs a new group, usually denoted $G \times H$ . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

In mathematics, a group is supersolvable if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.

In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S₆, the symmetric group on 6 elements.

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Glossary of group theory

Contents

A

C

D

E

F

G

H

I

L

N

O

P

Q

R

S

T

Basic definitions

Types of groups

See also

Related Research Articles