Higman group. Explicit examples, which turn out to be finitely presented, include the infinite Thompson groups T and V. Finitely presented torsion-free infinite simple groups were constructed by Burger and Mozes. Classification There is as yet no known classification for general (infinite) simple groups, and no such classification is expected. Finite simple groups 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: Zp – cyclic group of prime order An – alternating group for n ≥ 5 The alternating groups may be considered as groups of Lie type over the field with one element, which unites this family with the next, and thus all families of non-abelian finite simple groups may be considered to be of Lie type. One of 16 families of groups of Lie type The Tits group is generally considered of this form, though strictly speaking it is not of Lie type, but rather index 2 in a group of Lie type. One of 26 exceptions, the sporadic groups, of which 20 are subgroups or subquotients of the monster group and are referred to as the "Happy Family", while the remaining 6 are referred to as pariahs. Structure of finite simple groups 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. History for finite simple groups 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