of congruence classes modulo 3 (see modular arithmetic) is simple.
is not simple; the set of even integers is a non-trivial proper normal subgroup.
It is much more difficult to construct finitely generated infinite simple groups.
[5] Explicit examples, which turn out to be finitely presented, include the infinite Thompson groups
Finitely presented torsion-free infinite simple groups were constructed by Burger and Mozes.
[7] 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).
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.
There are two threads in the history of finite simple groups – the discovery and construction of specific simple groups and families, which took place from the work of Galois in the 1820s to the construction of the Monster in 1981; and proof that this list was complete, which began in the 19th century, most significantly took place 1955 through 1983 (when victory was initially declared), but was only generally agreed to be finished in 2004.
By 2018, its publication was envisioned as a series of 12 monographs,[8] the tenth of which was published in 2023.
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 are 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 group of a plane over a prime finite field, PSL(2,p), and remarked that they were simple for p not 2 or 3.
This is contained in his last letter to Chevalier,[10] and are the next example of finite simple groups.
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.
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.
The full classification is generally accepted as beginning with the Feit–Thompson theorem of 1962–1963 and being completed in 2004.
Soon after the construction of the Monster in 1981, a proof, totaling more than 10,000 pages, was supplied in 1983 by Daniel Gorenstein, that claimed to successfully list all finite simple groups.
This was premature, as gaps were later discovered in the classification of quasithin groups.
The gaps were filled in 2004 by a 1300 page classification of quasithin groups and the proof is now generally accepted as complete.
Sylow's 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 congruent 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[13] and, therefore, is not simple.
Burnside: A non-Abelian finite simple group has order divisible by at least three distinct primes.