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 Feitand Thompson ( 1963 ), where it was used to prove the Thompson uniqueness theorem.

Suppose that *G* is a finite group and *p* a prime such that all *p*-local subgroups are *p*-constrained. If *A* is a self-centralizing normal abelian subgroup of a *p*-Sylow subgroup such that *A* has rank at least 3, then the centralizer C_{G}(*A*) act transitively on the maximal *A*-invariant *q* subgroups of *G* for any prime *q* ≠ *p*.

In mathematics, the **classification of the finite simple groups** is a theorem stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. Group theory is central to many areas of pure and applied mathematics and the classification theorem has been called one of the great intellectual achievements of humanity. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

In mathematics, specifically group theory, given a prime number *p*, a ** p-group** is a group in which the order of every element is a power of

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 abstract algebra, a **finite group** is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups.

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, a **Frobenius group** is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius.

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.

The **Schur–Zassenhaus theorem** is a theorem in group theory which states that if is a finite group, and is a normal subgroup whose order is coprime to the order of the quotient group , then is a semidirect product of and . An alternative statement of the theorem is that any normal Hall subgroup of a finite group has a complement in . Moreover if either or is solvable then the Schur–Zassenhaus theorem also states that all complements of in G are conjugate. The assumption that either or is solvable can be dropped as it is always satisfied, but all known proofs of this require the use of the much harder Feit–Thompson theorem.

In mathematics, George Glauberman's **ZJ theorem** states that if a finite group *G* is *p*-constrained and *p*-stable and has a normal *p*-subgroup for some odd prime *p*, then *O*_{p′}(*G*)*Z*(*J* ) is a normal subgroup of *G*, for any Sylow *p*-subgroup*S*.

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 mathematical finite group theory, an **N-group** is a group all of whose local subgroups are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal 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, Thompson's original **uniqueness theorem** 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 mathematical finite group theory, the **Dade isometry** is an isometry from class function 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 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.

- 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 - Gorenstein, D. (1980),
*Finite groups*(2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR 0569209

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