- 211 · 33 · 52 · 13 = 17971200
- ≈ 2×107.
It is sometimes considered a 27th sporadic group.
History and properties
The Ree groups 2F4(22n+1) were constructed by Ree (1961), who showed that they are simple if n ≥ 1. The first member of this series 2F4(2) is not simple. It was studied by Jacques Tits (1964) who showed that it is almost simple, its derived subgroup 2F4(2)′ of index 2 being a new simple group, now called the Tits group. The group 2F4(2) is a group of Lie type and has a BN pair, but the Tits group itself does not have a BN pair. Because the Tits group is not strictly a group of Lie type, it is sometimes regarded as a 27th sporadic group.
The Tits group occurs as a maximal subgroup of the Fischer group Fi22. The groups 2F4(2) also occurs as a maximal subgroup of the Rudvalis group, as the point stabilizer of the rank-3 permutation action on 4060 = 1 + 1755 + 2304 points.
The Tits group is one of the simple N-groups, and was overlooked in John G. Thompson's first announcement of the classification of simple N-groups, as it had not been discovered at the time. It is also one of the thin finite groups.
L3(3):2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations.
2..5.4 Centralizer of an involution.
A6.22 (Two classes, fused by an outer automorphism)
The Tits group can be defined in terms of generators and relations by
- Parrott, David (1972), "A characterization of the Tits' simple group", Canadian Journal of Mathematics, 24: 672–685, doi:10.4153/cjm-1972-063-0, ISSN 0008-414X, MR 0325757
- Parrott, David (1973), "A characterization of the Ree groups 2F4(q)", Journal of Algebra, 27: 341–357, doi:10.1016/0021-8693(73)90109-9, ISSN 0021-8693, MR 0347965
- Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F4)", Bulletin of the American Mathematical Society, 67: 115–116, doi:10.1090/S0002-9904-1961-10527-2, ISSN 0002-9904, MR 0125155
- Stroth, Gernot (1980), "A general characterization of the Tits simple group", Journal of Algebra, 64 (1): 140–147, doi:10.1016/0021-8693(80)90138-6, ISSN 0021-8693, MR 0575787
- Tchakerian, Kerope B. (1986), "The maximal subgroups of the Tits simple group", Pliska Studia Mathematica Bulgarica, 8: 85–93, ISSN 0204-9805, MR 0866648
- Tits, Jacques (1964), "Algebraic and abstract simple groups", Annals of Mathematics, Second Series, 80: 313–329, doi:10.2307/1970394, ISSN 0003-486X, JSTOR 1970394, MR 0164968
- Wilson, Robert A. (1984), "The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits", Proceedings of the London Mathematical Society, Third Series, 48 (3): 533–563, doi:10.1112/plms/s3-48.3.533, ISSN 0024-6115, MR 0735227