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In mathematics, the term **Thompson group** or **Thompson's group** can refer to either

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

- The finite Thompson sporadic group
*Th*studied by John G. Thompson - The finite Thompson subgroup of a
*p*-group, the subgroup generated by the abelian subgroups of maximal order. - "Thompson subgroup" can also mean an analogue of the Weyl group used in the classical involution theorem
- The infinite Thompson groups
*F*,*T*and*V*studied by the logician Richard Thompson.

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

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 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.

Outside of mathematics, it may also refer to

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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.

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 the area of modern algebra known as group theory, the **Monster group** M (also known as the **Fischer–Griess Monster**, or the **Friendly Giant**) is the largest sporadic simple group, having order

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, more specifically in the field of group theory, a **solvable group** or **soluble group** is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

In algebraic geometry, an **algebraic group** is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.

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, the **Thompson groups** are three groups, commonly denoted
, which were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, *F* is the most widely studied, and is sometimes referred to as **the Thompson group** or **Thompson's group**.

**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, a **matrix group** is a group *G* consisting of invertible matrices over a specified field *K*, with the operation of matrix multiplication, and a **linear group** is an abstract group that is isomorphic to a matrix group over a field *K*, in other words, admitting a faithful, finite-dimensional representation over *K*.

In mathematics, a **polycyclic group** is a solvable group that satisfies the maximal condition on subgroups. Polycyclic groups are finitely presented, and this makes them interesting from a computational point of view.

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, in the field of group theory, a **locally finite group** is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.

In mathematics, the **Tits alternative**, named for Jacques Tits, is an important theorem about the structure of finitely generated linear 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 mathematics, a **Hecke algebra** can be one of several algebras, similar to the algebra of Hecke operators studied by Erich Hecke. The algebra of Hecke operators can be interpreted as an algebra of double cosets, and as a result the term "Hecke algebra" is also used for several similar algebras related to double cosets. In particular it can mean:

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.