Concept
Classification of finite simple groups
Related concepts (26)
Tits group
In group theory, the Tits group 2F4(2)′, named for Jacques Tits (tits), is a finite simple group of order 211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group. The Ree groups 2F4(22n+1) were constructed by , who showed that they are simple if n ≥ 1. The first member of this series 2F4(2) is not simple. It was studied by who showed that it is almost simple, its derived subgroup 2F4(2)′ of index 2 being a new simple group, now called the Tits group.
Composition series
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.
Quasithin group
In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non-trivial 2-subgroup of G.
Local analysis
In mathematics, the term local analysis has at least two meanings, both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a 'global' picture. These are forms of the approach. In group theory, local analysis was started by the Sylow theorems, which contain significant information about the structure of a finite group G for each prime number p dividing the order of G.
Fischer group
In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by . 3-transposition group The Fischer groups are named after Bernd Fischer who discovered them while investigating 3-transposition groups. These are groups G with the following properties: G is generated by a conjugacy class of elements of order 2, called 'Fischer transpositions' or 3-transpositions. The product of any two distinct transpositions has order 2 or 3.
Almost simple group
In mathematics, a group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group A is almost simple if there is a (non-abelian) simple group S such that Trivially, non-abelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group.