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In mathematical finite group theory, the **Thompson order formula**, introduced by John Griggs Thompson ( Held 1969 , p.279), 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 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 group theory, a branch of mathematics, the term *order* is used in two unrelated senses:

If a finite group *G* has exactly two conjugacy classes of involutions with representatives *t* and *z*, then the Thompson order formula ( Aschbacher 2000 , 45.6)( Suzuki 1986 , 5.1.7) states

- |G| = |C
_{G}(*z*)|*a*(*t*) + |C_{G}(*t*)|*a*(*z*)

Here *a*(*x*) is the number of pairs (*u*,*v*) with *u* conjugate to *t*, *v* conjugate to *z*, and *x* in the subgroup generated by *uv*.

Harris (1972 , 3.10) gives the following more complicated version of the Thompson order formula for the case when *G* has more than two conjugacy classes of involution.

where *t* and *z* are non-conjugate involutions, the sum is over a set of representatives *x* for the conjugacy classes of involutions, and *a*(*x*) is the number of ordered pairs of involutions *u*,*v* such that *u* is conjugate to *t*, *v* is conjugate to *z*, and *x* is the involution in the subgroup generated by *tz*.

The Thompson order formula can be rewritten as

where as before the sum is over a set of representatives *x* for the classes of involutions. The left hand side is the number of pairs on involutions (*u*,*v*) with *u* conjugate to *t*, *v* conjugate to *z*. The right hand side counts these pairs in classes, depending the class of the involution in the cyclic group generated by *uv*. The key point is that *uv* has even order (as if it had odd order then *u* and *v* would be conjugate) and so the group it generates contains a unique involution *x*.

In mathematics, especially group theory, the elements of any group may be partitioned into **conjugacy classes**; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. For an abelian group, each conjugacy class is a set containing one element.

In mathematics, the **Cayley–Dickson construction**, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as **Cayley–Dickson algebras**, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics.

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 number theory, the **local zeta function** is defined as

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

In mathematics, more specifically in group theory, the **character** of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

In group theory, a branch of abstract algebra, a **character table** is a two-dimensional table whose rows correspond to irreducible group representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation.

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

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, **Burnside theorem** in group theory states that if *G* is a finite group of order

In mathematics, the **Burnside ring** of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century, but the algebraic ring structure is a more recent development, due to Solomon (1967).

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, **Deligne–Lusztig theory** is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by Pierre Deligne and George Lusztig (1976).

In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of *S*_{6}, the symmetric group on 6 elements.

In mathematics, the **Arthur–Selberg trace formula** is a generalization of the Selberg trace formula from the group SL_{2} to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of *G*(**A**) on the discrete part *L*^{2}_{0}(*G*(*F*)∖*G*(**A**)) of *L*^{2}(*G*(*F*)∖*G*(**A**)) in terms of geometric data, where *G* is a reductive algebraic group defined over a global field *F* and **A** is the ring of adeles of *F*.

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 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 **Conway group***Co _{2}* is a sporadic simple group of order

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

- Aschbacher, Michael (2000),
*Finite group theory*, Cambridge Studies in Advanced Mathematics,**10**(2nd ed.), Cambridge University Press, ISBN 978-0-521-78675-1, MR 1777008 - Brauer, R.; Fowler, K. A. (1955), "On groups of even order",
*Annals of Mathematics*, Second Series,**62**: 565–583, doi:10.2307/1970080, ISSN 0003-486X, JSTOR 1970080, MR 0074414 - Harris, Morton E. (1972), "A characterization of odd order extensions of the finite projective symplectic groups PSp(4,q)",
*Transactions of the American Mathematical Society*,**163**: 311–327, doi:10.2307/1995724, ISSN 0002-9947, JSTOR 1995724, MR 0286897 - Held, Dieter (1969), "The simple groups related to M₂₄",
*Journal of Algebra*,**13**: 253–296, doi:10.1016/0021-8693(69)90074-X, ISSN 0021-8693, 0249500 - Suzuki, Michio (1986),
*Group theory. II*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],**248**, Berlin, New York: Springer-Verlag, ISBN 978-0-387-10916-9, MR 0815926

**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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