Simple group

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a simple group is a nontrivial

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, variously labeled e, 1, or 0, is the identity element. The group operation is e + e = e.Every trivial group is...

 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 only normal subgroup

Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group....

s are 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, variously labeled e, 1, or 0, is the identity element. The group operation is e + e = e.Every trivial group is...

 and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and 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...

, and the process can be repeated. If the group is finite

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...

, then eventually one arrives at uniquely determined simple groups by the Jordan–Hölder theorem.

For example, the cyclic group

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...

 G = Z/3Z of congruence classes modulo

Modulo operation

In computing, the modulo operation finds the remainder of division of one number by another.Given two numbers, and , a modulo n is the remainder, on division of a by n. For instance, the expression "7 mod 3" would evaluate to 1, while "9 mod 3" would evaluate to 0...

 3 (see modular arithmetic

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus...

) is simple.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a simple group is a nontrivial

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, variously labeled e, 1, or 0, is the identity element. The group operation is e + e = e.Every trivial group is...

 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 only normal subgroup

Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group....

s are 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, variously labeled e, 1, or 0, is the identity element. The group operation is e + e = e.Every trivial group is...

 and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and 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...

, and the process can be repeated. If the group is finite

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...

, then eventually one arrives at uniquely determined simple groups by the Jordan–Hölder theorem.

Examples

For example, the cyclic group

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...

 G = Z/3Z of congruence classes modulo

Modulo operation

In computing, the modulo operation finds the remainder of division of one number by another.Given two numbers, and , a modulo n is the remainder, on division of a by n. For instance, the expression "7 mod 3" would evaluate to 1, while "9 mod 3" would evaluate to 0...

 3 (see modular arithmetic

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus...

) is simple. If H is a subgroup of this group, its order (the number of elements) must be a divisor

Divisor

In mathematics, a divisor of an integer , also called a factor of , is an integer which evenly divides without leaving a remainder.-Explanation:...

 of the 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...

 of G which is 3. Since 3 is prime, its only divisors are 1 and 3, so either H is G, or H is the trivial group. On the other hand, the group G = Z/12Z is not simple. The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group

Abelian group

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...

 is normal. Similarly, the additive group Z of integer

Integer

The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....

s is not simple; the set of even integers is a non-trivial proper normal subgroup.

One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic group

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...

s of prime

Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...

 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 classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the alternating group

Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by A_n or Alt.For instance, the alternating group of degree 4...

 A₅ of order 60, and every simple group of order 60 is isomorphic

Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...

 to A₅. The second smallest nonabelian simple group is the projective special linear group PSL(2,7)

PSL(2,7)

In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane...

 of order 168, and it is possible to prove that every simple group of order 168 is isomorphic to PSL(2,7)

PSL(2,7)

In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane...

.

Classification

The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way prime number

Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...

s are the basic building blocks of the integer

Integer

The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....

s. This is expressed by the Jordan–Hölder theorem which states that any two 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...

 of a given group have the same length and the same factors, up to

Up to

In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one to which it is considered equivalent...

 permutation

Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the elements of a set to other elements of the same set, i.e., exchanging elements of a set.- Definitions :The general concept of permutation can be...

 and isomorphism

Isomorphism

In abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f ⁻¹ are homomorphisms, i.e., structure-preserving mappings....

. In a huge collaborative effort, the classification of finite simple groups

Classification of finite simple groups

The classification of the finite simple groups, also called the enormous theorem, is believed to classify all finite simple groups. 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 building blocks of the natural...

 was accomplished in 1982.

The famous theorem of Feit

Walter Feit

Walter Feit was a mathematician who worked in finite group theory and representation theory.He was born in Vienna and left for England in 1939. He moved to the United States in 1946 where he became an undergraduate at the University of Chicago. He did his Ph.D...

 and Thompson

John G. Thompson

John Griggs Thompson is a mathematician noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970 and the 2008 Abel Prize.-Biography:...

 states that every group of odd order is solvable

Solvable group

In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory...

. Therefore every finite simple group has even order unless it is cyclic of prime order.

Simple groups of infinite order also exist: simple Lie group

Simple Lie group

In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...

s and the infinite Thompson groups

Thompson groups

In mathematics, the Thompson groups are three groups, commonly denoted F, T and V, which were first studied by the logician Richard Thompson in 1965...

 T and V are examples of these.

The Schreier conjecture

Schreier conjecture

In finite group theory, the Schreier conjecture asserts that the group of outer automorphisms of every finite simple group is solvable. It was proposed by Otto Schreier, and is now known to be true as a result of the classification of finite simple groups, although , no simpler proof is known....

 asserts that the group of outer automorphisms of every finite simple group is solvable

Solvable group

In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory...

. This can be proved using the classification theorem.

Sporadic simple groups

In 1831 Évariste Galois

Évariste Galois

Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...

 discovered that the alternating group

Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by A_n or Alt.For instance, the alternating group of degree 4...

s on five or more points were simple. The next discoveries were by Camille Jordan

Camille Jordan

Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...

 in 1870. Jordan had found 4 families of simple matrix groups over finite field

Finite field

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

s of prime order. Later Jordan's results were generalized to arbitrary finite fields by Leonard Dickson. In the process he discovered several new infinite families of groups, now called the classical group

Classical group

The classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. There is a certain leeway in using the term classical group depending on the context. The term seems to have been coined by Hermann Weyl...

s. At about the same time, it was shown that a family of five groups, called the Mathieu group

Mathieu group

In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873...

s and first described by Émile Léonard Mathieu

Émile Léonard Mathieu

Émile Léonard Mathieu was a French mathematician. He is most famous for his work in group theory and mathematical physics. He has given his name to the Mathieu functions, Mathieu groups and Mathieu transformation...

 in 1861 and 1873, were also simple. Since these five groups were constructed by methods which did not yield infinitely many possibilities, they were called "sporadic

Sporadic group

In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...

" by William Burnside

William Burnside

William Burnside was an English mathematician. He is known mostly as an early contributor to the theory of finite groups....

 in his 1897 textbook. In 1981 Robert Griess announced that he had constructed Bernd Fischer

Bernd Fischer

Bernd Fischer is a German mathematician.He is best known to his contributions to the classification of finite simple groups, and three sporadic groups are named “Fischer groups” after him....

's "Monster group

Monster group

In the mathematical field of group theory, the Monster group M or F₁ is a group of finite order...

". The Monster is the largest sporadic simple group having order of 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. Each element of the Monster can be expressed as a 196,883 by 196,883 matrix. Soon after a proof, totaling more than 10,000 pages, was supplied that group theorists had successfully listed all finite simple groups.

Tests for nonsimplicity

Sylows' test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n.

Proof: If n is a prime-power, then a group of order n has a nontrivial center and, therefore, is not simple. If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.

Burnside: A non-Abelian finite simple group has order divisible by at least three distinct primes. This follows from Burnside's p-q theorem

Burnside theorem

In mathematics, Burnside's theorem in group theory states that if G is a finite group of orderwhere p and q are prime numbers, and a and b are non-negative integers, then G is solvable...

.