Glossary of group theory
Encyclopedia
A group is a set G closed
under a binary operation
• satisfying the following 3 axioms:
Basic examples for groups are the integers Z with addition operation, or rational numbers without zero Q\{0} with multiplication. More generally, for any ring
R, the units
in R form a multiplicative group
. See the group
article for an illustration of this definition and for further examples. Groups include, however, much more general structures than the above. Group theory is concerned with proving abstract statements about groups, regardless of the actual nature of element and the operation of the groups in question.
This glossary provides short explanations of some basic notions used throughout group theory. Please refer to group theory
for a general description of the topic. See also list of group theory topics.
H ⊂ G is a subgroup
if the restriction of • to H is a group operation on H. It is called normal
, if left and right coset
s agree, i.e. gH = Hg for all g in G. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup N in a group G naturally inherits a group structure, enabling the formation of the quotient group
, usually denoted G/N (also called a
factor group). The Butterfly lemma is a technical result on the lattice of subgroups
of a group.
Given a subset S of a group G, the smallest subgroup of G containing S is called the subgroup generated by S. It is often denoted <S>.
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
.
Given any set A, one can define a group as the smallest group containing the free semigroup
of A. This group consists of the finite strings called words that can be composed by elements from A and their inverses. Multiplication of strings is defined by concatenation, for instance
Every group G is basically a factor group of a free group generated by the set of its elements. This phenomenon is made formal with group presentations
.
The direct product
, direct sum, and semidirect product
of groups glue several groups together, in different ways. The direct product of a family of groups Gi, for example, is the cartesian product
of the sets underlying the various Gi, and the group operation is performed component-wise.
A group homomorphism
is a map f : G → H between two groups that preserves the structure imposed by the operation, i.e.
Bijective (in-, surjective) maps are isomorphism
s of groups (mono-
, epimorphism
s, respectively). The kernel ker(f) is always a normal subgroup of the group. For f as above, the fundamental theorem on homomorphisms
relates the structure of G and H, and of the kernel and image of the homomorphism, namely
One of the fundamental problems of group theory is the classification of groups up to
isomorphism.
Groups together with group homomorphisms form a category
.
In universal algebra
, groups are generally treated as algebraic structures of the form (G, •, e, −1), i.e. the identity element e and the map that takes every element a of the group to its inverse a−1 are treated as integral parts of the formal definition of a group.
. An important class is the group of permutations or symmetric group
s of N letters, denoted SN. Cayley's theorem
exhibits any finite group G as a subgroup of the symmetric group
on G. The theory of finite groups is very rich. Lagrange's theorem
states that the order of any subgroup H of a finite group G divides the order of G. A partial converse is given by the Sylow theorems: if pn is the greatest power of a prime
p dividing the order of a finite group G, then there exists a subgroup of order pn, and the number of these subgroups is also known. A projective limit of finite groups is called profinite. An important profinite group, fundamental for p-adic analysis
, class field theory
, and l-adic cohomology is the ring of p-adic integers and the profinite completion of Z, respectively
and 
Most of the facts from finite groups can be generalized directly to the profinite case.
Certain conditions on chains of subgroups
, parallel to the notion of Noetherian
and Artinian ring
s, allow to deduce further properties. For example the Krull-Schmidt theorem states that a group satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
Another, yet slightly weaker, level of finiteness is the following: a subset A of G is said to generate
the group if any element h can be written as the product of elements of A. A group is said to be finitely generated if it is possible to find a finite subset A generating the group. Finitely generated groups are in many respects as well-treatable as finite groups.
can be subdivided in several ways. A particularly well-understood class of groups are the so-called abelian
(in honor of Niels Abel, or commutative) groups, i.e. the ones satisfying
Another way of saying this is that the commutator
equals the identity element. A non-abelian group is a group that is not abelian. Even more particular, cyclic
groups are the groups generated
by a single element. Being either isomorphic to Z or to Zn, the integers modulo
n, they are always abelian. Any finitely generated abelian group
is known to be a direct sum of groups of these two types. The category of abelian groups is an abelian category
. In fact, abelian groups serve as the prototype of abelian categories. A converse is given by Mitchell's embedding theorem
.
(or derived group) is the subgroup generated by commutators [a, b], whereas the center is the subgroup of elements that commute with every other group element.
Given a group G and a normal subgroup N of G, denoted N ⊲ G, there is an exact sequence
:
where 1 denotes the trivial group
and H is the quotient G/N. This permits the decomposition of G into two smaller pieces. The other way round, given two groups N and H, a group G fitting into an exact sequence as above is called an extension of H by N. Given H and N there are many different group extensions G, which leads to the extension problem. There is always at least one extension, called the trivial extension, namely the direct sum G = N ⊕ H, but usually there are more. For example, the Klein four-group
is a non-trivial extension of Z2 by Z2. This is a first glimpse of homological algebra
and Ext functors
.
Many properties for groups, for example being a finite group
or a p-group
(i.e. the order of every element is a power of p) are stable under extensions and sub- and quotient groups, i.e. if N and H have the property, then so does G and vice versa. This kind of information is therefore preserved while breaking it into pieces by means of exact sequences. If this process has come to an end, i.e. if a group G does not have any (non-trivial) normal subgroups, G is called simple
. 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. The finite simple groups are known and classified
.
Repeatedly taking normal subgroups (if they exist) leads to normal series
:
i.e. any Gi is a normal subgroup of the next one Gi+1. A group is solvable
(or soluble) if it has a normal series all of whose quotients are abelian. Imposing further commutativity constraints on the quotients Gi+1 / Gi, one obtains central series
which lead to nilpotent group
s. They are an approximation of abelian groups in the sense that
for all choices of group elements gi.
There may be distinct normal series for a group G. If it is impossible to refine a given series by inserting further normal subgroups, it is called composition series
. By the Jordan–Hölder theorem any two composition series of a given group are equivalent.
, denoted by GL(n, F), is the group of
-by-
invertible matrices, where the elements of the matrices are taken from a field
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.
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
under a binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
• satisfying the following 3 axioms:
- AssociativityAssociativityIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
: For all a, b and c in G, (a • b) • c = a • (b • c). - Identity elementIdentity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
: There exists an e∈G such that for all a in G, e • a = a • e = a. - Inverse elementInverse elementIn abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
: For each a in G, there is an element b in G such that a • b = b • a = e, where e is an identity element.
Basic examples for groups are the integers Z with addition operation, or rational numbers without zero Q\{0} with multiplication. More generally, for any ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R, the units
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
in R form a multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
. See the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
article for an illustration of this definition and for further examples. Groups include, however, much more general structures than the above. Group theory is concerned with proving abstract statements about groups, regardless of the actual nature of element and the operation of the groups in question.
This glossary provides short explanations of some basic notions used throughout group theory. Please refer to group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
for a general description of the topic. See also list of group theory topics.
Basic definitions
A subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
H ⊂ G is a subgroup
Subgroup
In group theory, 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 x H is a group operation on H...
if the restriction of • to H is a group operation on H. It is called normal
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
, if left and right coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s agree, i.e. gH = Hg for all g in G. Normal subgroups play a distinguished role by virtue of the fact that the collection of cosets of a normal subgroup N in a group G naturally inherits a group structure, enabling the formation of the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
, usually denoted G/N (also called a
factor group). The Butterfly lemma is a technical result on the lattice of subgroups
Lattice of subgroups
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion....
of a group.
Given a subset S of a group G, the smallest subgroup of G containing S is called the subgroup generated by S. It is often denoted <S>.
Both subgroups and normal subgroups of a given group form a complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
under inclusion of subsets; this property and some related results are described by the lattice theorem
Lattice theorem
In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if N is a normal subgroup of a group G, then there exists a bijection from the set of all subgroups A of G such that A contains N, onto the set of all subgroups...
.
Given any set A, one can define a group as the smallest group containing the free semigroup
Free semigroup
In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences of zero or more elements from A. It is usually denoted A∗. The identity element is the unique sequence of zero elements, often called the empty string and denoted by ε or λ, and the...
of A. This group consists of the finite strings called words that can be composed by elements from A and their inverses. Multiplication of strings is defined by concatenation, for instance
Every group G is basically a factor group of a free group generated by the set of its elements. This phenomenon is made formal with group presentations
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
.
The direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
, direct sum, and semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of groups glue several groups together, in different ways. The direct product of a family of groups Gi, for example, is the cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the sets underlying the various Gi, and the group operation is performed component-wise.
A group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
is a map f : G → H between two groups that preserves the structure imposed by the operation, i.e.
- f(a•b) = f(a) • f(b).
Bijective (in-, surjective) maps are isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s of groups (mono-
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....
, epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
s, respectively). The kernel ker(f) is always a normal subgroup of the group. For f as above, the fundamental theorem on homomorphisms
Fundamental theorem on homomorphisms
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism....
relates the structure of G and H, and of the kernel and image of the homomorphism, namely
- G / ker(f) ≅ imImage (mathematics)In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
(f).
One of the fundamental problems of group theory is the classification of groups up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
isomorphism.
Groups together with group homomorphisms form a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
.
In universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, groups are generally treated as algebraic structures of the form (G, •, e, −1), i.e. the identity element e and the map that takes every element a of the group to its inverse a−1 are treated as integral parts of the formal definition of a group.
Finiteness conditions
The order |G| (or o(G)) of a group is the cardinality of G. If the order |G| is (in-)finite, then G itself is called (in-)finiteFinite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
. An important class is the group of permutations or symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
s of N letters, denoted SN. Cayley's theorem
Cayley's theorem
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
exhibits any finite group G as a subgroup of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on G. The theory of finite groups is very rich. Lagrange's theorem
Lagrange's theorem (group theory)
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange....
states that the order of any subgroup H of a finite group G divides the order of G. A partial converse is given by the Sylow theorems: if pn is the greatest power of a prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p dividing the order of a finite group G, then there exists a subgroup of order pn, and the number of these subgroups is also known. A projective limit of finite groups is called profinite. An important profinite group, fundamental for p-adic analysis
P-adic analysis
In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....
, class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
, and l-adic cohomology is the ring of p-adic integers and the profinite completion of Z, respectively
and Most of the facts from finite groups can be generalized directly to the profinite case.
Certain conditions on chains of subgroups
Ascending chain condition
The ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
, parallel to the notion of Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
and Artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
s, allow to deduce further properties. For example the Krull-Schmidt theorem states that a group satisfying certain finiteness conditions for chains of its subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
Another, yet slightly weaker, level of finiteness is the following: a subset A of G is said to generate
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
the group if any element h can be written as the product of elements of A. A group is said to be finitely generated if it is possible to find a finite subset A generating the group. Finitely generated groups are in many respects as well-treatable as finite groups.
Abelian groups
The category of groupsCategory of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
can be subdivided in several ways. A particularly well-understood class of groups are the so-called abelian
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
(in honor of Niels Abel, or commutative) groups, i.e. the ones satisfying
Another way of saying this is that the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
equals the identity element. A non-abelian group is a group that is not abelian. Even more particular, cyclic
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
groups are the groups generated
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
by a single element. Being either isomorphic to Z or to Zn, the integers modulo
Modulo
In the mathematical community, the word modulo is often used informally. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....
n, they are always abelian. Any finitely generated abelian group
Finitely generated abelian group
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the formwith integers n1,...,ns...
is known to be a direct sum of groups of these two types. The category of abelian groups is an abelian category
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
. In fact, abelian groups serve as the prototype of abelian categories. A converse is given by Mitchell's embedding theorem
Mitchell's embedding theorem
Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem, is a result stating that every abelian category admits a full and exact embedding into the category of R-modules...
.
Normal series
Most of the notions developed in group theory are designed to tackle non-abelian groups. There are several notions designed to measure how far a group is from being abelian. The commutator subgroupCommutator subgroup
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....
(or derived group) is the subgroup generated by commutators [a, b], whereas the center is the subgroup of elements that commute with every other group element.
Given a group G and a normal subgroup N of G, denoted N ⊲ G, there is an exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
:
- 1 → N → G → H → 1,
where 1 denotes the trivial group
Trivial group
In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...
and H is the quotient G/N. This permits the decomposition of G into two smaller pieces. The other way round, given two groups N and H, a group G fitting into an exact sequence as above is called an extension of H by N. Given H and N there are many different group extensions G, which leads to the extension problem. There is always at least one extension, called the trivial extension, namely the direct sum G = N ⊕ H, but usually there are more. For example, the Klein four-group
Klein four-group
In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
is a non-trivial extension of Z2 by Z2. This is a first glimpse of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
and Ext functors
Ext functor
In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...
.
Many properties for groups, for example being a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
or a p-group
P-group
In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
(i.e. the order of every element is a power of p) are stable under extensions and sub- and quotient groups, i.e. if N and H have the property, then so does G and vice versa. This kind of information is therefore preserved while breaking it into pieces by means of exact sequences. If this process has come to an end, i.e. if a group G does not have any (non-trivial) normal subgroups, G is called simple
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
. The name is misleading because a simple group can in fact be very complex. An example is the monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
, whose order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
is about 1054. The finite simple groups are known and classified
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
.
Repeatedly taking normal subgroups (if they exist) leads to normal series
Normal series
In mathematics, a subgroup series is a chain of subgroups:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G.Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups...
:
- 1 = G0 ⊲ G1 ⊲ ... ⊲ Gn = G,
i.e. any Gi is a normal subgroup of the next one Gi+1. A group is solvable
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
(or soluble) if it has a normal series all of whose quotients are abelian. Imposing further commutativity constraints on the quotients Gi+1 / Gi, one obtains central series
Central series
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial...
which lead to nilpotent group
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
s. They are an approximation of abelian groups in the sense that
- [...i>g1, g2], g3] ..., gn]=1
for all choices of group elements gi.
There may be distinct normal series for a group G. If it is impossible to refine a given series by inserting further normal subgroups, it is called composition series
Composition series
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...
. By the Jordan–Hölder theorem any two composition series of a given group are equivalent.
Other notions
General linear groupGeneral linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
, denoted by GL(n, F), is the group of
-by- invertible matrices, where the elements of the matrices are taken from a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
such as the real numbers or the complex numbers.Group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
(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
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
which is much easier to study.