In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a quasithin group is a
finite simple group 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 group, reductive linear algebraic group with values in a finite ...
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 mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...
.
More precisely it is a finite simple group of
characteristic 2 type and width 2. Here characteristic 2 type means that its
centralizers of
involutions resemble those of
groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an
abelian group of
odd order 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 of the algebraic group).
Classification
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the basis of which the classification of finite simple groups was announced as finished in 1983, was premature as the unpublished manuscript of his work was incomplete and contained serious gaps.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 characteristic are given by
*Groups of Lie type of characteristic 2 and rank 1 or 2, except that U
5(''q'') only occurs for ''q'' = 4
*PSL
4(2), PSL
5(2), Sp
6(2)
*The
alternating group
In mathematics, an alternating group is the Group (mathematics), 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 ...
s on 5, 6, 8, 9 points
*PSL
2(''p'') for ''p'' a
Fermat
Pierre de Fermat (; ; 17 August 1601 – 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 recognized for his d ...
or
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 1 ...
, L(3), L(3), G
2(3)
*The
Mathieu groups M
11, M
12, M
22, M
23, M
24, The
Janko groups J
2, J
3, J
4, the
Higman-Sims group, the
Held group, 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,
Richard Lyons, and
, then the only extra group that appears is the
Janko group J1.
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. ...
, 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