Tits group
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In group theory, the Tits group 2F4(2)′, named for Jacques Tits (French: [tits]), is a finite simple group of order
- 211 · 33 · 52 · 13 = 17,971,200.
This is the only simple group that is a derivative of a group of Lie type that is not strictly a group of Lie type in any series due to exceptional isomorphism. 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 2F4(2) of this series 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. The Tits group is member of the infinite family 2F4(22n+1)′ of commutator groups of the Ree groups, and thus by definition not sporadic. But because it is also 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 2F4(2).
The Tits group occurs as a maximal subgroup of the Fischer group Fi22. The group 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.
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:
L3(3):2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations.
2.[28].5.4 Centralizer of an involution.
L2(25)
22.[28].S3
A6.22 (Two classes, fused by an outer automorphism)
52:4A4
Presentation
The Tits group can be defined in terms of generators and relations by
- [math]\displaystyle{ a^2 = b^3 = (ab)^{13} = [a, b]^5 = [a, bab]^4 = ((ab)^4 ab^{-1})^6 = 1, \, }[/math]
where [a, b] is the commutator a−1b−1ab. It has an outer automorphism obtained by sending (a, b) to (a, b(ba)5b(ba)5).
Notes
- ↑ For instance, by the ATLAS of Finite Groups and its web-based descendant
References
- Parrott, David (1972), "A characterization of the Tits' simple group", Canadian Journal of Mathematics 24 (4): 672–685, doi:10.4153/cjm-1972-063-0, ISSN 0008-414X, https://books.google.com/books?id=TY5tZCQcK1IC&pg=PA672
- Parrott, David (1973), "A characterization of the Ree groups 2F4(q)", Journal of Algebra 27 (2): 341–357, doi:10.1016/0021-8693(73)90109-9, ISSN 0021-8693
- 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, http://www.ams.org/journals/bull/1961-67-01/S0002-9904-1961-10527-2/home.html
- 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
- Tchakerian, Kerope B. (1986), "The maximal subgroups of the Tits simple group", Pliska Studia Mathematica Bulgarica 8: 85–93, ISSN 0204-9805
- Tits, Jacques (1964), "Algebraic and abstract simple groups", Annals of Mathematics, Second Series 80 (2): 313–329, doi:10.2307/1970394, ISSN 0003-486X
- 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
External links
de:Gruppe vom Lie-Typ#Die Tits-Gruppe
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