Thin group (finite group theory)

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.

Janko (1972) defined thin groups and classified those of characteristic 2 type in which all 2-local subgroups are solvable. The thin simple groups were classified by Aschbacher (1976, 1978). The list of finite simple thin groups consists of:

• The projective special linear groups PSL₂(q)
• The projective special linear groups PSL₃(p) for p = 1 + 2^a or p = 1 + 2^a3, and PSL₃(4)
• The projective special unitary groups PSU₃(p) for p = 1 - 2^a or p = 1 - 2^a3, and PSU₃(2ⁿ)
• The Suzuki groups Sz(2ⁿ)
• The Tits group ²F₄(2)'
• The Steinberg group ³D₄(2)
• The Mathieu group M₁₁
• The Janko group J1