Concepts in group theory  
category of groups  
subgroups, normal subgroups  
group homomorphisms, kernel, image, quotient  
direct product, direct sum  
semidirect product, wreath product  
Types of groups  

simple, finite, infinite  
discrete, continuous  
multiplicative, additive  
cyclic, abelian, dihedral  
nilpotent, solvable  
list of group theory topics  
glossary of group theory 
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. If the group is finite, then eventually one arrives at uniquely determined simple groups by the Jordan–Hölder theorem.
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For example, the cyclic group G = Z/3Z of congruence classes modulo 3 (see modular arithmetic) is simple. If H is a subgroup of this group, its order (the number of elements) must be a divisor of the order 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 is normal. Similarly, the additive group Z of integers is not simple; the set of even integers is a nontrivial 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 groups of prime order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the alternating group A_{5} of order 60, and every simple group of order 60 is isomorphic to A_{5}. The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and it is possible to prove that every simple group of order 168 is isomorphic to PSL(2,7).
Simple groups of infinite order also exist: simple Lie groups and the infinite Thompson groups T and V are examples of these.
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 numbers are the basic building blocks of the integers. This is expressed by the Jordan–Hölder theorem which states that any two composition series of a given group have the same length and the same factors, up to permutation and isomorphism. In a huge collaborative effort, the classification of finite simple groups was declared accomplished in 1983 by Daniel Gorenstein, though some problems surfaced (specifically in the classification of quasithin groups, which were plugged in 2004) and some doubts have been raised about the validity of such a large proof.
Briefly, finite simple groups are classified as lying in one of 18 families, or being one of 26 exceptions:
The famous theorem of Feit and Thompson states that every group of odd order is solvable. Therefore every finite simple group has even order unless it is cyclic of prime order.
The Schreier conjecture asserts that the group of outer automorphisms of every finite simple group is solvable. This can be proved using the classification theorem.
Simple groups have been studied at least since early Galois theory, where Évariste Galois realized that the fact that the alternating groups on five or more points was simple (and hence not solvable), which he proved in 1831, was the reason that one could not solve the quintic in radicals. Galois also constructed the projective special linear groups over prime finite fields, PSL(2,p), which are the next example of finite simple groups.^{[1]}
The next discoveries were by Camille Jordan in 1870.^{[2]} Jordan had found 4 families of simple matrix groups over finite fields of prime order, which are now known as the classical groups.
At about the same time, it was shown that a family of five groups, called the Mathieu groups and first described by Émile Léonard Mathieu 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" by William Burnside in his 1897 textbook.
Later Jordan's results on classical groups were generalized to arbitrary finite fields by Leonard Dickson, following the classification of complex simple Lie algebras by Wilhelm Killing. Dickson also constructed exception groups of type G2 and E_{6} as well, but not of types F4, E7, or E8 (Wilson 2009, p. 2). In the 1950s the work on groups of Lie type was continued, with Claude Chevalley giving a uniform construction of the classical groups and the groups of exceptional type in a 1955 paper. This omitted certain known groups (the projective unitary groups), which were obtained by "twisting" the Chevalley construction. The remaining groups of Lie type were produced by Steinberg, Tits, and Herzig (who produced ^{3}D_{4}(q) and ^{2}E_{6}(q)) and by Suzuki and Ree (the Suzuki–Ree groups).
These groups (the groups of Lie type, together with the cyclic groups, alternating groups, and the five exceptional Mathieu groups) were believed to be a complete list, but after a lull of almost a century since the work of Mathieu, in 1964 the first Janko group was discovered, and the remaining 20 sporadic groups were discovered or conjectured in 1965–1975, culminating in 1981, when Robert Griess announced that he had constructed Bernd Fischer's "Monster group". 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. Some gaps were later discovered, notably in the classification of quasithin groups, which were eventually replaced in 2004 by a 1,300 page classification of quasithin groups.
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 primepower, then a group of order n has a nontrivial center^{[3]} 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 psubgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow psubgroup is unique, and therefore it is normal. Since it is a proper, nonidentity subgroup, the group is not simple.
Burnside: A nonAbelian finite simple group has order divisible by at least three distinct primes. This follows from Burnside's pq theorem.
