mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a quasithin group is a

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

that resembles a

group of Lie type 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 phra ...

of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its

centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...

s of involutions resemble those of

groups of Lie type 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 phra ...

over fields of characteristic 2, and the width is roughly the maximal rank of an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

normalizing a non-trivial 2-subgroup of ''G''. When ''G'' is a group of Lie type of characteristic 2 type, the width is usually the rank (the dimension of a

maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...

of the algebraic group).

Classification

The classification of quasithin groups is a crucial part of the

classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory 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 i ...

. The quasithin groups were classified in a 1221-page paper by . An earlier announcement by of the classification, on the basis of which the classification of finite simple groups was announced as finished in 1983, was premature as the unpublished manuscript of his work was incomplete and contained serious gaps. According to , the finite simple quasithin groups of

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...

characteristic are given by *Groups of Lie type of characteristic 2 and rank 1 or 2, except that U₅(''q'') only occurs for ''q'' = 4 *PSL₄(2), PSL₅(2), Sp₆(2) *The

alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...

s on 5, 6, 8, 9, points *PSL₂(''p'') for ''p'' a

Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...

, L(3), L(3), G₂(3) *The Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄, The Janko groups J₂, J₃, J₄, the Higman-Sims group, the

Held group In the area of modern algebra known as group theory, the Held group ''He'' is a sporadic simple group of order : 21033527317 = 4030387200 : ≈ 4. History ''He'' is one of the 26 sporadic groups and was found by during an inv ...

, and the Rudvalis group. If the condition "even characteristic" is relaxed to "even type" in the sense of the revision of the classification by

Daniel Gorenstein Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissert ...

, Richard Lyons, and

Ronald Solomon Ronald "Ron“ Mark Solomon (b. 15 December 1948Mathematicians who classified finite groups

, then the only extra group that appears is the

Janko group J1 In the area of modern algebra known as group theory, the Janko group ''J1'' is a sporadic simple group of order : 233571119 = 175560 : ≈ 2. History ''J1'' is one of the 26 sporadic groups and was originally described by Zvo ...

References

* * * * (unpublished typescript) *{{citation, url=https://www.ams.org/bull/2006-43-01/S0273-0979-05-01071-2 , last=Solomon, first = Ronald, authorlink=Ronald Solomon, journal=

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...

, volume= 43 , year=2006, pages= 115–121 , title= Review of The classification of quasithin groups. I, II by Aschbacher and Smith , doi=10.1090/s0273-0979-05-01071-2, doi-access=free Finite groups