Related Research Articles
In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic. 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 the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by Issai Schur in his work on projective representations.
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.
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, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the partial ordering being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.
In the area of modern algebra known as group theory, the Janko groupJ4 is a sporadic simple group of order
In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. The classification of quasithin groups is a crucial part of 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 ∩ 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 group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Aschbacher for rank 3 and by Gorenstein & Lyons (1983) for rank at least 4. The three classes are groups of GF(2) type, groups of "standard type" for some odd prime, and groups of uniqueness type, where Aschbacher proved that there are no simple groups.
In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number p, the Sylow p-subgroups of the 2-local subgroups are cyclic. Informally, these are the groups that resemble rank 1 groups of Lie type over a finite field of characteristic 2.
In mathematical finite group theory, the Thompson order formula, introduced by John Griggs Thompson, 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 finite group theory, a branch of mathematics, a block, sometimes called Aschbacher block, is a subgroup giving an obstruction to Thompson factorization and pushing up. Blocks were introduced by Michael Aschbacher.
In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem.
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 Fischer groupFi24 or F24′ or F3+ is a sporadic simple group of order
In the area of modern algebra known as group theory, the Fischer groupFi23 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Fischer groupFi22 is a sporadic simple group of order
Jan Saxl was a Czech-British mathematician, and a professor at the University of Cambridge. He was known for his work in finite group theory, particularly on consequences of the classification of finite simple groups.
References
- Aschbacher, Michael (2000), Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10 (2nd ed.), Cambridge University Press, ISBN 0-521-78145-0, MR 1777008
- Aschbacher, Michael (1981), "On the failure of the Thompson factorization in 2-constrained groups", Proceedings of the London Mathematical Society, Third Series, 43 (3): 425–449, doi:10.1112/plms/s3-43.3.425, ISSN 0024-6115, MR 0635564
- Thompson, John G. (1966), "Factorizations of p-solvable groups", Pacific Journal of Mathematics , 16 (2): 371–372, doi: 10.2140/pjm.1966.16.371 , ISSN 0030-8730, MR 0188296
 | This group theory-related article is a stub. You can help Wikipedia by expanding it. |