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 + 2^a3^b and PSL₃(4)
• The projective special unitary groups PSU₃(p) for p =−1 + 2^a3^b 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

See also

• Quasithin group

References

• 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