Glossary of group theory

abelian group

A group ${\displaystyle (G,*)}$ is abelian if ${\displaystyle *}$ is commutative, i.e. ${\displaystyle g*h=h*g}$ for all ${\displaystyle g}$,${\displaystyle h}$ ∈ ${\displaystyle G}$. Likewise, a group is nonabelian if this relation fails to hold for any pair ${\displaystyle g}$,${\displaystyle h}$ ∈ ${\displaystyle G}$.

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 ∈ G such that hg = gh for all g ∈ 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.

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⁻¹h⁻¹gh. Some authors define the commutator as [g, h] = ghg⁻¹h⁻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

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

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.

conjugate elements

Two elements x and y of a group G are conjugate if there exists an element g ∈ G such that g⁻¹xg = y. The element g⁻¹xg, 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 ^gx. Conjugacy is an equivalence relation. Its equivalence classes are called conjugacy classes.

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.

factor group

See quotient group.

finite group

A finite group is a group of finite order, that is, a group with a finite number of elements.

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.

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

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

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:

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

with

${\displaystyle A_{i}\triangleleft G}$.

normal subgroup

A subgroup N of a group G is normal in G (denoted ${\displaystyle N\triangleleft G}$) if the conjugation of an element n of N by an element g of G is always in N (gng⁻¹ ∈ N for all g ∈ G and n ∈ N). A normal subgroup N of a group G can be used to construct the quotient group G/N (G mod N).

order of a group

The order of a group ${\displaystyle (G,*)}$ is the cardinality (i.e. number of elements) of ${\displaystyle G}$. A group with finite order is called a finite group.

order of a group element

The order of an element ${\displaystyle g}$ of a group is the smallest positive integer ${\displaystyle n}$ such that ${\displaystyle g^{n}=e.}$ If no such integer exists, then the order of ${\displaystyle g}$ is said to be infinite. The order of a finite group is divisible by the order of every element.

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.

p-group

If ${\displaystyle p}$ is prime, then a ${\displaystyle p}$-group is one in which the order of every element is a power of ${\displaystyle p}$. A finite group is a p-group if and only if the order of the group is a power of ${\displaystyle p}$.

p-subgroup

A subgroup which is also a ${\displaystyle p}$-group. The study of ${\displaystyle p}$-subgroups is the central object of the Sylow theorems.

quotient group

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

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. 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.

simple group

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

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:

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

Subgroup. A subset ${\displaystyle H}$ of a group ${\displaystyle (G,*)}$ which remains a group when the operation ${\displaystyle *}$ is restricted to ${\displaystyle H}$ is called a subgroup of ${\displaystyle G}$.

Given a subset ${\displaystyle S}$ of ${\displaystyle G}$. We denote by ${\displaystyle <S>}$ the smallest subgroup of ${\displaystyle G}$ containing ${\displaystyle S}$. ${\displaystyle <S>}$ is called the subgroup of ${\displaystyle G}$ generated by ${\displaystyle S}$.

Normal subgroup. ${\displaystyle H}$ is a normal subgroup of ${\displaystyle G}$ if for all ${\displaystyle g}$ in ${\displaystyle G}$ and ${\displaystyle h}$ in ${\displaystyle H}$, ${\displaystyle g*h*g^{-1}}$also belongs to ${\displaystyle H}$.

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.

Group homomorphism. These are functions ${\displaystyle f\colon (G,*)\to (H,\times )}$ that have the special property that

${\displaystyle f(a*b)=f(a)\times f(b),}$

for any elements ${\displaystyle a}$ and ${\displaystyle b}$ of ${\displaystyle G}$.

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.

Group isomorphism. Group homomorphisms that have inverse functions. The inverse of an isomorphism, it turns out, must also be a homomorphism.

Isomorphic groups. 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. One of the fundamental problems of group theory is the classification of groups up to isomorphism.

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.

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

Simple group. Simple groups are those groups having only ${\displaystyle e}$ 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 ${\displaystyle A}$, one can define a group as the smallest group containing the free semigroup of ${\displaystyle A}$. The group consists of the finite strings (words) that can be composed by elements from ${\displaystyle A}$, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance ${\displaystyle (abb)*(bca)=abbbca.}$

Every group ${\displaystyle (G,*)}$ is basically a factor group of a free group generated by ${\displaystyle G}$. Please 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 ${\displaystyle n}$-by-${\displaystyle n}$ invertible matrices, where the elements of the matrices are taken from a field ${\displaystyle F}$ 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.

• Glossary of Lie groups and Lie algebras
• Glossary of ring theory
• Composition series
• Normal series