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. The Monster has a faithful 196,883-dimensional representation in the 196,884-dimensional Griess algebra, meaning that each element of the Monster can be expressed as a 196,883 by 196,883 matrix. Classification The full classification is generally accepted as starting with the Feit–Thompson theorem of 1962–63, largely lasting until 1983, but only being finished in 2004. Soon after the construction of the Monster in 1981, a proof, totaling more than 10,000 pages, was supplied that group theorists had successfully listed all finite simple groups, with victory declared in 1983 by Daniel Gorenstein. This was premature – 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, which is now generally accepted as complete. Tests for nonsimplicity 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 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 theorem. See also Almost simple group Characteristically simple group Quasisimple group Semisimple group List of finite simple groups References Notes Textbooks Papers Properties of groups