In mathematics, a quasithin group is roughly a finite simple group of characteristic 2 type

Characteristic 2 type

In mathematical finite group theory, a group is said to be of characteristic 2 type or even type or of even characteristic if it resembles a group of Lie type over a field of characteristic 2....

 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 classification of quasithin groups is a crucial part of the classification of finite simple groups

Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

. The quasithin groups were classified in a 1221 page paper by . An earlier announcement by of the classification, on which basis the classification of finite simple groups was announced as finished in 1983, was premature as the unpublished manuscript of his work was incomplete and contained serious gaps.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 U₅(q) only occurs for q=4.
• PSL₄(2), PSL₅(2), Sp₆(2)
• The alternating groups on 5, 6, 8, 9, points.
• PSL₂(p) for p a Fermat or Mersenne prime, L(3), L(3), G₂(3)
• The Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄, The Janko group
  Janko group
  In mathematics, a Janko group is one of the four sporadic simple groups named for Zvonimir Janko. Janko constructed the first Janko group J1 in 1965. At the same time, Janko also predicted the existence of J2 and J3. In 1976, he suggested the existence of J4...
  
  s J₂, J₃, J₄, the Higman-Sims group
  Higman-Sims group
  In the mathematical field of group theory, the Higman–Sims group HS is a sporadic simple group found by of orderThe Higman–Sims group was discovered in 1967, when Higman and Sims were attending a presentation by Marshall Hall on the Hall–Janko group...
  
  , the Held group
  Held group
  In the mathematical field of group theory, the Held group He is one of the 26 sporadic simple groups, and has order...
  
  , and the Rudvalis group
  Rudvalis group
  In the mathematical field of group theory, the Rudvalis group Ru is a sporadic simple group of order-Properties:The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer the Ree group...
  
  .

If the condition "even characteristic" is relaxed to "even type" in the sense of the Gorenstein-Lyons-Solomon revision of the classification, then the only extra group that appears is the Janko group J1

Janko group J1

In mathematics, the smallest Janko group, J1, is a simple sporadic group of order 175560. It was originally described by Zvonimir Janko and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century...

.