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In mathematical finite group theory, a **Thompson factorization**, introduced by Thompson ( 1966 ), is an expression of some finite groups as a product of two subgroups, usually normalizers or centralizers of *p*-subgroups for some prime *p*.

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

In abstract algebra, a **finite group** is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, an element of the set. In the case of a finite group, the set is finite.

In mathematics and abstract algebra, **group theory** studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In mathematics, the **classification of the finite simple groups** is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

In mathematical group theory, the **Schur multiplier** or **Schur multiplicator** is the second homology group of a group *G*. It was introduced by Issai Schur (1904) in his work on projective representations.

In mathematics, the **Feit–Thompson theorem**, or **odd order theorem**, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson.

In mathematics, the **lattice of subgroups** of a group is the lattice whose elements are the subgroups of , with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.

In mathematics, in the field of group theory, a **component** of a finite group is a quasisimple subnormal subgroup. Any two distinct components commute. The product of all the components is the layer of the group.

In mathematical finite group theory, the **Thompson subgroup** of a finite *p*-group *P* refers to one of several characteristic subgroups of *P*. John G. Thompson (1964) originally defined to be the subgroup generated by the abelian subgroups of *P* of maximal rank. More often the Thompson subgroup is defined to be the subgroup generated by the abelian subgroups of *P* of maximal order or the subgroup generated by the elementary abelian subgroups of *P* of maximal rank. In general these three subgroups can be different, though they are all called the Thompson subgroup and denoted by .

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*. When *G* is a group of Lie type of characteristic 2 type, the width is usually the rank.

In mathematics, a **signalizer functor** gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The **signalizer functor theorem** gives conditions under which a signalizer functor comes from a subgroup. The idea is to try to construct a -subgroup of a finite group , which has a good chance of being normal in , by taking as generators certain -subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian -subgroups of The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including Gorenstein (1969) who defined signalizer functors, Glauberman (1976) who proved the Solvable Signalizer Functor Theorem for solvable groups, and McBride who proved it for all groups. This theorem is needed to prove the so-called "dichotomy" stating that a given nonabelian finite simple group either has local characteristic two, or is of component type. It thus plays a major role in the classification of finite simple groups.

In finite group theory, an area of abstract algebra, a **strongly embedded subgroup** of a finite group *G* is a proper subgroup *H* of even order such that *H* ∩ *H*^{g} has odd order whenever *g* is not in *H*. The **Bender–Suzuki theorem**, proved by Bender (1971) extending work of Suzuki (1962, 1964), classifies the groups *G* with a strongly embedded subgroup *H*. It states that either

*G*has cyclic or generalized quaternion Sylow 2-subgroups and*H*contains the centralizer of an involution- or
*G*/*O*(*G*) has a normal subgroup of odd index isomorphic to one of the simple groups PSL_{2}(*q*), Sz(*q*) or PSU_{3}(*q*) where*q*≥4 is a power of 2 and*H*is*O*(*G*)N_{G}(*S*) for some Sylow 2-subgroup*S*.

In mathematical group theory, a **C-group** is a group such that the centralizer of any involution has a normal Sylow 2-subgroup. They include as special cases **CIT-groups** where the centralizer of any involution is a 2-group, and **TI-groups** where any Sylow 2-subgroups have trivial intersection.

In mathematical finite group theory, the **trichotomy theorem** divides the simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Aschbacher for rank 3 and by Gorenstein & Lyons (1983) for rank at least 4. The three classes are groups of GF(2) type, groups of "standard type" for some odd prime, and groups of uniqueness type, where Aschbacher proved that there are no simple groups.

In the mathematical classification of finite simple groups, the **component theorem** of Aschbacher shows that if *G* is a simple group of odd type, and various other assumptions are satisfied, then *G* has a centralizer of an involution with a "standard component" with small centralizer.

In the mathematical classification of finite simple groups, a **thin group** is a finite group such that for every odd prime number *p*, the Sylow *p*-subgroups of the 2-local subgroups are cyclic. Informally, these are the groups that resemble rank 1 groups of Lie type over a finite field of characteristic 2.

In mathematical finite group theory, the **Thompson order formula**, introduced by John Griggs Thompson, gives a formula for the order of a finite group in terms of the centralizers of involutions, extending the results of Brauer & Fowler (1955).

In mathematical finite group theory, a **block**, sometimes called **Aschbacher block**, is a subgroup giving an obstruction to Thompson factorization and pushing up. Blocks were introduced by Michael Aschbacher.

In mathematical finite group theory, the **Dade isometry** is an isometry from class functions on a subgroup *H* with support on a subset *K* of *H* to class functions on a group *G*. It was introduced by Dade (1964) as a generalization and simplification of an isometry used by Feit & Thompson (1963) in their proof of the odd order theorem, and was used by Peterfalvi (2000) in his revision of the character theory of the odd order theorem.

In mathematical finite group theory, the **uniqueness case** is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem.

In mathematical finite group theory, the **classical involution theorem** of Aschbacher classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mostly groups of Lie type over a field of odd characteristic. Berkman (2001) extended the classical involution theorem to groups of finite Morley rank.

In the area of modern algebra known as group theory, the **Fischer group***Fi _{24}* or F

- Aschbacher, Michael (2000),
*Finite group theory*, Cambridge Studies in Advanced Mathematics,**10**(2nd ed.), Cambridge University Press, ISBN 0-521-78145-0, MR 1777008 - Aschbacher, Michael (1981), "On the failure of the Thompson factorization in 2-constrained groups",
*Proceedings of the London Mathematical Society*, Third Series,**43**(3): 425–449, doi:10.1112/plms/s3-43.3.425, ISSN 0024-6115, MR 0635564 - Thompson, John G. (1966), "Factorizations of
*p*-solvable groups",*Pacific Journal of Mathematics*,**16**(2): 371–372, doi:10.2140/pjm.1966.16.371, ISSN 0030-8730, MR 0188296

**Michael George Aschbacher** is an American mathematician best known for his work on finite groups. He was a leading figure in the completion of the classification of finite simple groups in the 1970s and 1980s. It later turned out that the classification was incomplete, because the case of quasithin groups had not been finished. This gap was fixed by Aschbacher and Stephen D. Smith in 2004, in a pair of books comprising about 1300 pages. Aschbacher is currently the Shaler Arthur Hanisch Professor of Mathematics at the California Institute of Technology.

**Cambridge University Press** (**CUP**) is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world. It also holds letters patent as the Queen's Printer.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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