Quasithin Group - Classification

Classification

The classification of quasithin groups is a crucial part of the classification of finite simple groups. The quasithin groups were classified in a 1221 page paper by Aschbacher and Smith (2004, 2004b). An earlier announcement by Mason (1980) of the classification, on which basis the classification of finite simple groups was announced as finished in 1983, was premature as the unpublished manuscript (Mason 1981) of his work was incomplete and contained serious gaps.

According to Aschbacher & Smith (2004b, theorem 0.1.1), 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, Lε
  3(3), Lε
  4(3), G₂(3)
• The Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄, The Janko groups J₂, J₃, J₄, 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 Gorenstein-Lyons-Solomon revision of the classification, then the only extra group that appears is the Janko group J1.