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 p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.
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.
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, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups.
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, 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.
Michio Suzuki was a Japanese mathematician who studied group theory.
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.
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:
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 ∩ Hg 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 PSL2(q), Sz(q) or PSU3(q) where q≥4 is a power of 2 and H is O(G)NG(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 the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22n+1), 2B2(22n+1), Suz(22n+1), or G(22n+1), form an infinite family of groups of Lie type found by Suzuki (1960), that are simple for n ≥ 1. These simple groups are the only finite non-abelian ones with orders not divisible by 3.
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.
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.