Finite group

# Finite group

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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a finite group is a 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...

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
Local analysis
In mathematics, the term local analysis has at least two meanings - both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a 'global' picture...

of finite groups, and the theory of solvable group
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...

s and 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. A complete determination of the structure of all finite groups is too much to hope for; the number of possible structures soon becomes overwhelming. However, the complete classification of the finite simple groups
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...

was achieved, meaning that the "building blocks" from which all finite groups can be built are now known, as each finite group has a 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...

.

During the second half of the twentieth century, mathematicians such as Chevalley
Claude Chevalley
Claude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...

and Steinberg
Robert Steinberg
Robert Steinberg is a mathematician at the University of California, Los Angeles who invented the Steinberg representation, the Steinberg group in algebraic K-theory, and the Steinberg groups in Lie theory that yield finite simple groups over finite fields. He received his Ph.D...

also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear group
General 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...

s over finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s.
Finite groups often occur when considering symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

of mathematical or
physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s,
which may be viewed as dealing with "continuous symmetry
Continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...

", is strongly influenced by the associated Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

s. These are finite groups generated by reflections which act on a finite dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. The properties of finite groups can thus play a role in subjects such as theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

.

## Number of groups of a given order

Given a positive integer n, it is not at all a routine matter to determine how many 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...

types of groups of order n there are. Every group of 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...

order is 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...

, since 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....

implies that the cyclic subgroup generated by
any of its non-identity elements is the whole group.
If n is the square of a prime, then there are exactly two possible isomorphism types of group of order n, both of which are abelian. If n is a higher power of a prime, then results of Graham Higman
Graham Higman
Graham Higman FRS was a leading British mathematician. He is known for his contributions to group theory....

and Charles Sims
Charles Sims (mathematician)
Charles Coffin Sims is an American mathematician best known for his work in group theory. Together with Donald G. Higman he discovered the Higman–Sims group, one of the sporadic groups...

give asymptotically correct estimates for the number of isomorphism types of groups of order n, and the number grows very rapidly as the power increases.

Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclic when q < p are primes with p-1 not divisible by q. For a necessary and sufficient condition, see cyclic number
Cyclic number (group theory)
A cyclic number is a natural number n such that n and φ are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic iff any group of order n is cyclic.Any prime number is clearly cyclic...

.

If n is squarefree, then any group of order n is solvable. A theorem of William Burnside
William Burnside
William Burnside was an English mathematician. He is known mostly as an early contributor to the theory of finite groups....

, proved using group characters
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....

, states that every group of order n is solvable when n is divisible by fewer than three distinct primes. By the Feit–Thompson theorem
Feit–Thompson theorem
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...

, which has a long and complicated proof, every group of order n is solvable when n is odd.

For every positive integer n, most groups of order n are 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...

. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the classification of finite simple groups
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...

. For any positive integer n there are at most two simple groups of order n, and there are infinitely many positive integers n for which there are two non-isomorphic simple groups of order n.

### Table of distinct groups of order n

Order n # Groups Abelian Non-Abelian
1 1 1 0
2 1 1 0
3 1 1 0
4 2 2 0
5 1 1 0
6 2 1 1
7 1 1 0
8 5 3 2
9 2 2 0
10 2 1 1
11 1 1 0
12 5 2 3
13 1 1 0
14 2 1 1
15 1 1 0
16 14 5 9
17 1 1 0
18 5 2 3
19 1 1 0
20 5 2 3
21 2 1 1
22 2 1 1
23 1 1 0
24 15 3 12
25 2 2 0

• Association scheme
Association scheme
The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics,...

• Classification of finite simple groups
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...

• List of finite simple groups
• 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....

• Cauchy's theorem (group theory)
Cauchy's theorem (group theory)
Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy. It states that if G is a finite group and p is a prime number dividing the order of G , then G contains an element of order p...

• Abelian group
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...

• Non-abelian group
• 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...

• List of small groups
• Representation theory of finite groups
Representation theory of finite groups
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...

• Modular representation theory
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...

• Monstrous moonshine
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...

• Profinite group
• Infinite group theory
• Finite ring
Finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements....

## External references

• Number of groups of order n
• A classifier for groups of small order