**Classification of the Finite Simple Groups**

$Basic History$

The classification of the finite simple groups is also called the enormous theorem, and is believed to classify all finite simple groups. Galois was the first one to discover these groups in 1830. The finite simple groups all fit into a table much like a periodic table of elements. But there are also groups that do not fit into the table. The groups that fit the table are called Lie type groups and the ones that do not fit in the table are called Sporadic type groups.

$Simple Group$

A simple group in mathematics is a nontrivial group whose only normaly subgroups are the trivial group and the group itself. A normal subgroup exists if the left and right cosets of that subgroup are equal. An example of a simple group would be: $A_5$ or $Z_p$ where p is a prime number.

$Classification Theorem$

$Lie Type$

Finite simple groups of Lie type form most of the nonabelian finite simple groups. There are four different types of Lie type groups. These different types include: Classical Groups, Chevalley Groups, Steinberg Groups, and Suzuki-Ree Groups.

$Classical Group$ Classical group is a special linear, orthogonal, symplectic, or unitary group.

The Chevalley Groups are an integral form for all the complex simple Lie algebras.

The Steinberg Groups are reconstructions of Chevalleys group with some modifications on the unitary groups which Chevalley had omitted.

Suzuki found a new infinite series of groups that seemed unrelated to the known algebraic groups.

$Sporadic Groups$

The Sporadic Groups are the 26 exceptional groups that do not follow a systematic pattern with in the finite simple groups. Five of the sporadic groups were founded by Mathieu in the 1860's. The other 21 of these groups were founded between 1965 and 1975 by many other mathematicians.

$The Monster$

The Monster Group is the largest group of the Sporadic group. The Monster Group contains 20 out of the 26 Sporadic groups.

The group was predicted in 1973 by Bernd Fischer and Robert Griess; it was first constructed in 1980 by Robert Griess, but not proven until 1992 by Richard Borcherds. It is a group of finite order 246 • 320 • 59 • 76 • 112 • 133 • 17 • 19 • 23 • 29 • 31 • 41 • 47 • 59 • 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 • 1053. Even with its size it is a group that has no normal subgroups and it is the largest sporadic group. It also contains 20 of the 26 other sporadic groups. Smallest # of dimensions which the Monster can act nontrivially = 196,883 and its characteristic table is equal to 194x194. It also contains at least 43 conjugacy classes of maximal subgroups. There are two 196882 by 196882 matrices that together generate the monster group.

http://en.wikipedia.org/wiki/File:Finitesubgroups.svg

The Monster group also describes the optimal packing of 24-Dimension Hyperballs and is very common in Physics on the research of String Theory. String Theory is a developing physicis theory that attempts toreconcile quantum mechanics and general relativity.

http://en.wikipedia.org/wiki/File:String_theory.svg

The prime divisors of Monster group's order are exactly the 15 supersingular primes: 2,3,5,7,11,13,17,19,23,29,31,41,47,59,& 71 which arise in the context of elliptic curves.

http://en.wikipedia.org/wiki/File:EllipticCurveCatalog.svg

Monster Group Atlas:

http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M/

$Doubts about the Proof$

There were doubts whether the proof of this theorem is complete and correct since it is spread out over 500 different articles. These doubts were justified when gaps in the agrument were found. Although to date all gaps in the argument have been filled, but some parts of the proof have yet to be published. One of the early gaps was the unpublished classification of quasithin groups. But there now exists a proof lasting around 1300 pages long on the quasithin group that is published.

$Examples$

Alternating Groups

The alternating group on $n$ symbols is the set of all even permutations of $S_n$ the symmetric group on $n$ symbols. It is usually denoted by $A_n$. This is a normal subgroup of $S_n$ namely the kernel of the homomorphism that sends every even permutation to 1 and the odd permutations to −1. Because every permutation is either even or odd, and there is a bijection between the two (multiply every even permutation by a transposition), the index of $A_n$ in $S_N$ is 2 $A_3$ is simple because it only has three elements, and the simplicity of $A_n$ for $n\geq 5$ can be proved by an elementary argument. The simplicity of the alternating groups is an important fact that Évariste Galois required in order to prove the insolubility by radicals of the general polynomial of degree higher than four.

References:

http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups

http://www.math.uic.edu/~ronan/The_Classification

http://planetmath.org/encyclopedia/ExamplesOfFiniteSimpleGroups.html

http://en.wikipedia.org/wiki/Monster_group

http://en.wikipedia.org/wiki/String_theory