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In mathematical finite group theory, Thompson's original **uniqueness theorem**( Feit & Thompson 1963 , theorems 24.5 and 25.2) states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3. Bender (1973) gave a shorter proof of the uniqueness theorem.

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, a **simple group** is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.

In mathematics, a **group** is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

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.

**John Griggs Thompson** is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize.

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, in the realm of group theory, a group is said to be a **CA-group** or **centralizer abelian group** if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theorem and the classification of finite simple groups. Several important infinite groups are CA-groups, such as free groups, Tarski monsters, and some Burnside groups, and the locally finite CA-groups have been classified explicitly. CA-groups are also called **commutative-transitive groups** because commutativity is a transitive relation amongst the non-identity elements of a group if and only if the group is a CA-group.

In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of : are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are solvable. Further progress was made showing that **CN-groups**, groups in which the centralizer of a non-identity element is nilpotent, of odd order are solvable. The complete solution was given in, but further work on CN-groups was done in, giving more detailed information about the structure of these groups. For instance, a non-solvable CN-group *G* is such that its largest solvable normal subgroup *O*_{∞}(*G*) is a 2-group, and the quotient is a group of even order.

In mathematics, the **Feit–Thompson conjecture** is a conjecture in number theory, suggested by Walter Feit and John G. Thompson (1962). The conjecture states that there are no distinct prime numbers *p* and *q* such that

In group theory, **Bender's method** is a method introduced by Bender (1970) for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2-subgroups Bender (1970b), and Gorenstein and Walter's classification of groups with dihedral Sylow 2-subgroups. Bender's method involves studying a maximal subgroup *M* containing the centralizer of an involution, and its generalized Fitting subgroup *F*^{*}(*M*).

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 mathematics, the **Walter theorem**, proved by John H. Walter, describes the finite groups whose Sylow 2-subgroup is abelian. Bender (1970) used Bender's method to give a simpler proof.

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.

**Everett Clarence Dade** is a mathematician at University of Illinois at Urbana–Champaign working on finite groups and representation theory, who introduced the Dade isometry and Dade's conjecture.

In finite group theory, a **p-stable group** for an odd prime *p* is a finite group satisfying a technical condition introduced by Gorenstein and Walter in order to extend Thompson's uniqueness results in the odd order theorem to groups with dihedral Sylow 2-subgroups.

In mathematical finite group theory, the **Thompson transitivity theorem** gives conditions under which the centralizer of an abelian subgroup *A* acts transitively on certain subgroups normalized by *A*. It originated in the proof of the odd order theorem by Feit and Thompson (1963), where it was used to prove the Thompson uniqueness theorem.

In mathematics, a **3-step group** is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different.

In mathematical representation theory, **coherence** is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. The general notion of coherence was developed by Feit, as a generalization of the proof by Frobenius of the existence of a Frobenius kernel of a Frobenius group and of the work of Brauer and Suzuki on exceptional characters. Feit & Thompson developed coherence further in the proof of the Feit–Thompson theorem that all groups of odd order are solvable.

In mathematical finite group theory, the **Puig subgroup**, introduced by Puig (1976), is a characteristic subgroup of a *p*-group analogous to the Thompson subgroup.

In mathematical group theory, the **Thompson replacement theorem** is a theorem about the existence of certain abelian subgroups of a *p*-group. The **Glauberman replacement theorem** is a generalization of it introduced by Glauberman.

- Bender, Helmut (1970), "On the uniqueness theorem",
*Illinois Journal of Mathematics*,**14**: 376–384, ISSN 0019-2082, MR 0262351 - Bender, Helmut; Glauberman, George (1994),
*Local analysis for the odd order theorem*, London Mathematical Society Lecture Note Series,**188**, Cambridge University Press, ISBN 978-0-521-45716-3, MR 1311244 - Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order",
*Pacific Journal of Mathematics*,**13**: 775–1029, ISSN 0030-8730, MR 0166261

An **International Standard Serial Number** (**ISSN**) is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is especially helpful in distinguishing between serials with the same title. ISSN are used in ordering, cataloging, interlibrary loans, and other practices in connection with serial literature.

* Mathematical Reviews* is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of

**George Glauberman** is a mathematician at the University of Chicago who works on finite simple groups. He proved the ZJ theorem and the Z^{*} theorem.

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