George Glauberman | |
---|---|

Born | George Isaac Glauberman March 3, 1941 New York City, New York, US |

Academic background | |

Alma mater | |

Doctoral advisor | R. H. Bruck |

Academic work | |

Discipline | Mathematics |

Institutions | University of Chicago |

Doctoral students | |

Main interests | Finite simple groups |

**George Isaac Glauberman** (born 1941) is a mathematician at the University of Chicago who works on finite simple groups. He proved the ZJ theorem and the Z^{*} theorem.

Born in New York City on March 3, 1941, Glauberman did his undergraduate studies at the Polytechnic Institute of Brooklyn, graduating in 1961, and earned a master's degree from Harvard University in 1962.^{ [1] } He obtained his PhD degree from the University of Wisconsin–Madison in 1965, under the supervision of Richard Bruck.^{ [2] } He has had 22 PhD students, including Ahmed Chalabi and Peter Landrock, the president and founder of Cryptomathic. He has co-authored with J. L. Alperin, Simon P. Norton, and Zvi Arad.

In 1970 he was an invited speaker at the International Congress of Mathematicians at Nice. In 2012 he became a fellow of the American Mathematical Society.^{ [3] }

- Glauberman, George (1964), "On loops of odd order",
*Journal of Algebra*,**1**: 374–396, doi: 10.1016/0021-8693(64)90017-1 , ISSN 0021-8693, MR 0175991 - Glauberman, George (1966), "Central elements in core-free groups",
*Journal of Algebra*,**4**: 403–420, doi: 10.1016/0021-8693(66)90030-5 , ISSN 0021-8693, MR 0202822, Zbl 0145.02802 - Glauberman, George (1968), "A characteristic subgroup of a p-stable group",
*Canadian Journal of Mathematics*,**20**: 1101–1135, doi:10.4153/cjm-1968-107-2, ISSN 0008-414X, MR 0230807 - Glauberman, George (1968), "Correspondences of characters for relatively prime operator groups.",
*Canadian Journal of Mathematics*,**20**: 1465–1488, doi:10.4153/cjm-1968-148-x, ISSN 0008-414X, MR 0232866 - Glauberman, George (1968), "On loops of odd order. II",
*Journal of Algebra*,**8**: 393–414, doi: 10.1016/0021-8693(68)90050-1 , ISSN 0021-8693, MR 0222198 - 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

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

**Joachim "Jim" Lambek** was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.D. degree in 1950 with Hans Zassenhaus as advisor.

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 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, the **Brauer–Suzuki theorem**, proved by Brauer & Suzuki (1959), Suzuki (1962), Brauer (1964), states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a centre of order 2. In particular, such a group cannot be simple.

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, 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 mathematics, especially in the area of algebra known as group theory, the term **Z-group** refers to a number of distinct types of groups:

**Stephen Arthur Jennings** was a mathematician who made contributions to the study of modular representation theory. His advisor was Richard Brauer, and his student Rimhak Ree discovered two infinite series of finite simple groups known as the Ree groups. Jennings was an editor of Mathematics Magazine and an acting president of the University of Victoria.

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 the mathematical theory of automorphic forms, the **fundamental lemma** relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups. It was conjectured by Robert Langlands (1983) in the course of developing the Langlands program. The fundamental lemma was proved by Gérard Laumon and Ngô Bảo Châu in the case of unitary groups and then by Ngô (2010) for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of Lie algebras. *Time* magazine placed Ngô's proof on the list of the "Top 10 scientific discoveries of 2009". In 2010, Ngô was awarded the Fields medal for this proof.

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 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 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 **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 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.

**Richard Hubert Bruck** was an American mathematician best known for his work in the field of algebra, especially in its relation to projective geometry and combinatorics.

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.

- ↑ Curriculum vitae, retrieved 2013-07-21.
- ↑ George Glauberman at the Mathematics Genealogy Project
- ↑ List of Fellows of the American Mathematical Society, retrieved 2013-01-19.

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