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) and PSL3(p) for p = 1 + 2a3b and PSL3(4)
- The projective special unitary groups PSU3(p) for p =−1 + 2a3b and b = 0 or 1 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
- Aschbacher, Michael (1976), "Thin finite simple groups", Bulletin of the American Mathematical Society, 82 (3): 484, doi:10.1090/S0002-9904-1976-14063-3, ISSN 0002-9904, MR 0396735
- Aschbacher, Michael (1978), "Thin finite simple groups", Journal of Algebra, 54 (1): 50–152, doi:10.1016/0021-8693(78)90022-4, ISSN 0021-8693, MR 0511458
- Janko, Zvonimir (1972), "Nonsolvable finite groups all of whose 2-local subgroups are solvable. I", Journal of Algebra, 21: 458–517, doi:10.1016/0021-8693(72)90009-9, ISSN 0021-8693, MR 0357584