Tits group
In group theory, the Tits group ^{2}F_{4}(2)′, named for Jacques Tits (French: [tits]), is a finite simple group of order
 2^{11} · 3^{3} · 5^{2} · 13 = 17971200
 ≈ 2×10^{7}.
Algebraic structure → Group theory Group theory 



Infinite dimensional Lie group

It is sometimes considered a 27th sporadic group.
History and properties
The Ree groups ^{2}F_{4}(2^{2n+1}) were constructed by Ree (1961), who showed that they are simple if n ≥ 1. The first member of this series ^{2}F_{4}(2) is not simple. It was studied by Jacques Tits (1964) who showed that it is almost simple, its derived subgroup ^{2}F_{4}(2)′ of index 2 being a new simple group, now called the Tits group. The group ^{2}F_{4}(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.[1]
The Schur multiplier of the Tits group is trivial and its outer automorphism group has order 2, with the full automorphism group being the group ^{2}F_{4}(2).
The Tits group occurs as a maximal subgroup of the Fischer group Fi_{22}. The groups ^{2}F_{4}(2) also occurs as a maximal subgroup of the Rudvalis group, as the point stabilizer of the rank3 permutation action on 4060 = 1 + 1755 + 2304 points.
The Tits group is one of the simple Ngroups, and was overlooked in John G. Thompson's first announcement of the classification of simple Ngroups, as it had not been discovered at the time. It is also one of the thin finite groups.
The Tits group was characterized in various ways by Parrott (1972, 1973) and Stroth (1980).
Maximal subgroups
Wilson (1984) and Tchakerian (1986) independently found the 8 classes of maximal subgroups of the Tits group as follows:
L_{3}(3):2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations.
2.[2^{8}].5.4 Centralizer of an involution.
L_{2}(25)
2^{2}.[2^{8}].S_{3}
A_{6}.2^{2} (Two classes, fused by an outer automorphism)
5^{2}:4A_{4}
Presentation
The Tits group can be defined in terms of generators and relations by
where [a, b] is the commutator a^{−1}b^{−1}ab. It has an outer automorphism obtained by sending (a, b) to (a, b(ba)^{5}b(ba)^{5})
Notes
 For instance, by the ATLAS of Finite Groups and its webbased descendant
References
 Parrott, David (1972), "A characterization of the Tits' simple group", Canadian Journal of Mathematics, 24: 672–685, doi:10.4153/cjm19720630, ISSN 0008414X, MR 0325757
 Parrott, David (1973), "A characterization of the Ree groups ^{2}F_{4}(q)", Journal of Algebra, 27: 341–357, doi:10.1016/00218693(73)901099, ISSN 00218693, MR 0347965
 Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F_{4})", Bulletin of the American Mathematical Society, 67: 115–116, doi:10.1090/S000299041961105272, ISSN 00029904, MR 0125155
 Stroth, Gernot (1980), "A general characterization of the Tits simple group", Journal of Algebra, 64 (1): 140–147, doi:10.1016/00218693(80)901386, ISSN 00218693, MR 0575787
 Tchakerian, Kerope B. (1986), "The maximal subgroups of the Tits simple group", Pliska Studia Mathematica Bulgarica, 8: 85–93, ISSN 02049805, MR 0866648
 Tits, Jacques (1964), "Algebraic and abstract simple groups", Annals of Mathematics, Second Series, 80: 313–329, doi:10.2307/1970394, ISSN 0003486X, 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/s348.3.533, ISSN 00246115, MR 0735227