In

_{''n''} on a _{''n''} is ''n''!.

_{''n''} is a group all of whose elements are powers of a particular element ''a'' where , the identity. A typical realization of this group is as the complex roots of unity. Sending ''a'' to a

group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

closely related to the group ''G''(''k'') of rational points of a reductive

group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

''G'' is

group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

of odd order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

is solvable. It was proved by

classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...

is a theorem stating that every

order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

''n'' there are. Every group of classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...

. 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''.

GroupNames

*

for groups of small order {{Authority control Properties of groups

abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

, a finite group is a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

whose underlying set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

s and permutation group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s.
The study of finite groups has been an integral part of group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ernő Rubik has bee ...

since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...

(those with no nontrivial normal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

) was completed in 2004.
History

During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially thelocal theory
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of finite groups and the theory of solvable and nilpotent group
In mathematics, specifically group theory, a nilpotent group ''G'' is a Group (mathematics), group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series termina ...

s. As a consequence, the complete classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...

was achieved, meaning that all those simple group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s from which all finite groups can be built are now known.
During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg
Steinberg Media Technologies GmbH (trading as Steinberg) is a German musical software and hardware company based in Hamburg with satellite offices in Siegburg and London. It develops music writing, recording, arranging, and editing software, most ...

also increased our understanding of finite analogs of classical groups
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and other related groups. One such family of groups is the family of general linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s over finite field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s.
Finite groups often occur when considering symmetry
Symmetry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie group
In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...

s, which may be viewed as dealing with "continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant un ...

", is strongly influenced by the associated Weyl group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

. The properties of finite groups can thus play a role in subjects such as theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in contrast to experimental ph ...

and chemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
A fact is an occurrence in the real world. T ...

.
Examples

Permutation groups

The symmetric group Sfinite set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of ''n'' symbols is the group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

whose elements are all the permutations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the ''n'' symbols, and whose group operation
form the Rubik's Cube group.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...

is the composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

of such permutations, which are treated as from the set of symbols to itself. Since there are ''n''! (''n'' factorial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

) possible permutations of a set of ''n'' symbols, it follows that the order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

(the number of elements) of the symmetric group SCyclic groups

A cyclic group Zprimitive root of unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...

gives an isomorphism between the two. This can be done with any finite cyclic group.
Finite abelian groups

Anabelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, also called a commutative group, is a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

). They are named after Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

.
An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group
In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...

of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of Germa ...

and Ludwig Stickelberger
Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a Swiss
Swiss may refer to:
* the adjectival form of Switzerland
*Swiss people
Places
*Swiss, Missouri
*Swiss, North Carolina
*Swiss, West Virginia
*Swiss, Wisconsin
Other uses
*Swi ...

and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

.
Groups of Lie type

A group of Lie type is alinear algebraic group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

''G'' with values in the field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

''k''. Finite groups of Lie type give the bulk of nonabelian finite simple groups. Special cases include the classical groups
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.
Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclic
Cycle or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in social scienc ...

, symmetric
Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

and alternating groups, with the projective special linear group
In mathematics, especially in the group theory, group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced Group action (mathematics), action of the general linear group ...

s over prime finite fields, PSL(2, ''p'') being constructed by Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ...

in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated a ...

's theorem that the projective special linear group
In mathematics, especially in the group theory, group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced Group action (mathematics), action of the general linear group ...

PSL(2, ''q'') is simple for ''q'' ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(''n'', ''q'') of finite simple groups. Other classical groups were studied by Leonard Dickson
Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as ...

in the beginning of 20th century. In the 1950s Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...

realized that after an appropriate reformulation, many theorems about semisimple Lie group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s admit analogues for algebraic groups over an arbitrary field ''k'', leading to construction of what are now called ''Chevalley groups''. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (''Tits simplicity theorem''). Although it was known since 19th century that other finite simple groups exist (for example, Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the sporadic groups
In group theory, a sporadic group is one of the 26 exceptional Group (mathematics), groups found in the classification of finite simple groups.
A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group an ...

, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their ''geometry'' in the sense of Tits.
The belief has now become a theorem – the classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...

. Inspection of the list of finite simple groups shows that groups of Lie type over a finite field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

include all the finite simple groups other than the cyclic groups, the alternating groups, the Tits group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by E ...

, and the 26 sporadic simple group
In group theory, a sporadic group is one of the 26 exceptional Group (mathematics), groups found in the classification of finite simple groups.
A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group an ...

s.
Main theorems

Lagrange's theorem

For any finite group ''G'', theorder
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

(number of elements) of every subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...

''H'' of ''G'' divides the order of ''G''. The theorem is named after Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaArthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

, states that every isomorphic
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

to a subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...

of the symmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

acting on ''G''. This can be understood as an example of the group action
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of ''G'' on the elements of ''G''.
Burnside's theorem

Burnside's theorem ingroup theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ernő Rubik has bee ...

states that if ''G'' is a finite group of order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

''p'q'', where ''p'' and ''q'' are prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s, and ''a'' and ''b'' are non-negative
In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third ...

integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, then ''G'' is solvable. Hence each
non-Abelian finite simple group
In mathematics, the classification of finite simple groups states that every Finite group, finite simple group is cyclic group, cyclic, or alternating group, alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
...

has order divisible by at least three distinct primes.
Feit–Thompson theorem

The Feit–Thompson theorem, or odd order theorem, states that every finiteClassification of finite simple groups

Thefinite simple group
In mathematics, the classification of finite simple groups states that every Finite group, finite simple group is cyclic group, cyclic, or alternating group, alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
...

belongs to one of the following families:
* A cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

with prime order;
* An alternating group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of degree at least 5;
* A simple group of Lie type;
* One of the 26 sporadic simple group
In group theory, a sporadic group is one of the 26 exceptional Group (mathematics), groups found in the classification of finite simple groups.
A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group an ...

s;
* The Tits group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by E ...

(sometimes considered as a 27th sporadic group).
The finite simple groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s are the basic building blocks of the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s. The Jordan–Hölder theoremIn abstract algebra, a composition series provides a way to break up an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

is a more precise way of stating this fact about finite groups. However, a significant difference with respect to the case of integer factorization
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...

is that such "building blocks" do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same composition series In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...

or, put in another way, the extension problem does not have a unique solution.
The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. (d.1992), Lyons
Lyon or Lyons (, , ; frp, Liyon, ) is the third-largest city and second-largest urban area of France. It is located at the confluence of the rivers Rhône
The Rhône ( , ) is a major river in France and Switzerland, arising in the Alps ...

, and Solomon
Solomon (; he, , ), ''Šlēmūn''; : سُلَيْمَان ', also : ' or '; el, Σολομών ''Solomōn''; : Salomon) also called Jedidiah (, ), was, according to the and Christian , a fabulously wealthy and wise monarch of the who suc ...

are gradually publishing a simplified and revised version of the proof.
Number of groups of a given order

Given a positive integer ''n'', it is not at all a routine matter to determine how manyisomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

types of groups of prime
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

order is cyclic
Cycle or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in social scienc ...

, because Lagrange's theorem 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 Fellow of the Royal Society, FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory.
Biography
Higman was born in Louth, Lincolnshire and attended Sutton High Sc ...

and Charles Sims 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
Peter Ludwig Mejdell Sylow () (12 December 1832 – 7 September 1918) was a Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic ...

. For example, every group of order ''pq'' is cyclic when are primes with not divisible by ''q''. For a necessary and sufficient condition, see cyclic number.
If ''n'' is squarefree
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, then any group of order ''n'' is solvable. Burnside's theorem
William Burnside.
In mathematics, Burnside's theorem in group theory states that if ''G'' is a finite group of Order (group theory), order p^a q^b where ''p'' and ''q'' are prime numbers, and ''a'' and ''b'' are negative and positive numbers, non-n ...

, proved using group characters, states that every group of order ''n'' is solvable when ''n'' is divisible by fewer than three distinct primes, i.e. if , where ''p'' and ''q'' are prime numbers, and ''a'' and ''b'' are non-negative integers. By the Feit–Thompson theoremIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, 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. 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 Table of distinct groups of order ''n''

See also

*Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every List of finite simple groups, finite simple group is either cyclic groups, cyclic, or alternating groups, alternating, or it belongs to a broad infinite ...

* Association scheme
*List of finite simple groups
In mathematics, the classification of finite simple groups states that every Finite group, finite simple group is cyclic group, cyclic, or alternating group, alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. ...

*Cauchy's theorem (group theory)Several theorems are named after Augustin-Louis Cauchy
Baron
Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the tit ...

*P-group
In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a Group (mathematics), group in which the Order of a group element, order of every element is a Power (mathematics), power of ''p''. That is, for each element ...

* List of small groups
*Representation theory of finite groups
The representation theory of group (mathematics), groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on Group action (mathematics), operations of groups on vector spaces. Nevertheless, ...

*Modular representation theory
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field (mathematics), field ''K'' of positive characteristic (algebra), characteristic ''p ...

*Monstrous moonshine
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

*Profinite groupIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

*Finite ringIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Commuting probability
*Finite State Machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation
A model is an informative representation of an object, person or system. ...

References

Further reading

*External links

* * * * Small groups oGroupNames

*

for groups of small order {{Authority control Properties of groups