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In mathematical finite group theory, the **Thompson subgroup***J*(*P*) of a finite *p*-group *P* refers to one of several characteristic subgroups of *P*. Thompson (1964) originally defined *J*(*P*) to be the subgroup generated by the abelian subgroups of *P* of maximal rank. More often the Thompson subgroup *J*(*P*) 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 *J*(*P*).

In group theory, a branch of mathematics, given a group *G* under a binary operation ∗, a subset *H* of *G* is called a **subgroup** of *G* if *H* also forms a group under the operation ∗. More precisely, *H* is a subgroup of *G* if the restriction of ∗ to *H* × *H* is a group operation on *H*. This is usually denoted *H* ≤ *G*, read as "*H* is a subgroup of *G*".

In abstract algebra, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after early 19th century mathematician Niels Henrik Abel.

In the mathematical subject of group theory, the **rank of a group***G*, denoted rank(*G*), can refer to the smallest cardinality of a generating set for *G*, that is

- Glauberman normal p-complement theorem
- ZJ theorem
- Puig subgroup, a subgroup analogous to the Thompson subgroup

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 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 the area of modern algebra known as group theory, the **Conway groups** are the three sporadic simple groups Co_{1}, Co_{2} and Co_{3} along with the related finite group Co_{0} introduced by (Conway 1968, 1969).

In the area of modern algebra known as group theory, the **Tits group**^{2}*F*_{4}(2)′, named for Jacques Tits (French: [tits]), is a finite simple group of order

In the area of modern algebra known as group theory, the **Thompson group***Th* is a sporadic simple group of order

In mathematics, a **Ree group** is a group of Lie type over a finite field constructed by Ree from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.

In mathematics, in the area of abstract algebra known as group theory, an **A-group** is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known about their structure.

In mathematics, George Glauberman's **Z* theorem** is stated as follows:

Z* theorem:LetGbe a finite group, withO(G) being its maximal normal subgroup of odd order. IfTis a Sylow 2-subgroup ofGcontaining an involution not conjugate inGto any other element ofT, then the involution lies inZ*(G), which is the inverse image inGof the center ofG/O(G).

In mathematics, the **Tits alternative**, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.

In mathematics, a **p-constrained group** is a finite group resembling the centralizer of an element of prime order *p* in a group of Lie type over a finite field of characteristic *p*. They were introduced by Gorenstein and Walter (1964, p.169) in order to extend some of Thompson's results about odd groups to groups with dihedral Sylow 2-subgroups.

In mathematical group theory, a **normal p-complement** of a finite group for a prime *p* is a normal subgroup of order coprime to *p* and index a power of *p*. In other words the group is a semidirect product of the normal *p*-complement and any Sylow *p*-subgroup. A group is called **p-nilpotent** if it has a normal *p*-complement.

In abstract algebra, the **focal subgroup theorem** describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to. The focal subgroup theorem relates the ideas of transfer and fusion such as described in. Various applications of these ideas include local criteria for *p*-nilpotence and various non-simplicity criterion focussing on showing that a finite group has a normal subgroup of index *p*.

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 **Gorenstein–Walter theorem**, proved by Gorenstein and Walter (1965a, 1965b, 1965c), states that if a finite group *G* has a dihedral Sylow 2-subgroup, and *O*(*G*) is the maximal normal subgroup of odd order, then *G*/*O*(*G*) is isomorphic to a 2-group, or the alternating group *A*_{7}, or a subgroup of PΓL_{2}(*q*) containing PSL_{2}(*q*) for *q* an odd prime power. Note that A_{5} ≈ PSL_{2}(4) ≈ PSL_{2}(5) and A_{6} ≈ PSL_{2}(9).

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 **Dempwolff group** is a finite group of order 319979520 = 2^{15}·3^{2}·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group as the subgroup fixing a certain lattice in the Lie algebra of , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

In mathematics, **Alvis–Curtis duality** is a duality operation on the characters of a reductive group over a finite field, introduced by Charles W. Curtis (1980) and studied by his student Dean Alvis (1979). Kawanaka introduced a similar duality operation for Lie algebras.

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.

**John Harris Walter** is an American mathematician known for proving the Walter theorem in the theory of finite groups.

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.

- Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1996),
*The classification of the finite simple groups. Number 2. Part I. Chapter G*, Mathematical Surveys and Monographs,**40**, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0390-5, MR 1358135 - Thompson, John G. (1964), "Normal p-complements for finite groups",
*Journal of Algebra*,**1**: 43–46, doi:10.1016/0021-8693(64)90006-7, ISSN 0021-8693, MR 0167521 - Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture",
*Journal of Algebra*,**13**: 149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR 0245683

**Daniel E. Gorenstein** was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation a duality principle for plane curves that motivated Grothendieck's introduction of Gorenstein rings. He was a major influence on the classification of finite simple groups.

The **American Mathematical Society** (**AMS**) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

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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