Tits group

In mathematics, the Tits group ²F₄(2)′ is a finite simple group of order 17971200 = 2¹¹ · 3³ · 5² · 13 found by Jacques Tits (1964).

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 its derived subgroup ²F₄(2)′ of index 2 was a new simple group. The group ²F₄(2) is a group of Lie type and has a BN pair, but the Tits group does not, so is strictly speaking not a group of Lie type, though it is usually classed with the groups of Lie type in lists of simple groups as it is so close to one.

Properties

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 group ²F₄(2) 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.

Wilson (1984) and Tchakerian (1986) independently found the 8 classes of maximal subgroups of the Tits group.

The Tits group is one of the simple N-groups, and was overlooked in John 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).

Presentation

The Tits group can be defined in terms of generators and relations by

$a^2 = b^3 = (ab)^{13} = [a, b]^5 = [a, bab]^4 = (ababababab^{-1})^6 = 1, \,$

where [a, b] is the commutator. It has an outer automorphism obtained by sending (a, b) to (a, bbabababababbababababa).

References

• Parrott, David (1972), "A characterization of the Tits' simple group", Canadian Journal of Mathematics 24: 672–685, ISSN 0008-414X, MR0325757, http://books.google.com/books?id=TY5tZCQcK1IC&pg=PA672 
• 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, MR0347965 
• 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, MR0125155, 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, MR575787 
• Tchakerian, Kerope B. (1986), "The maximal subgroups of the Tits simple group", Pliska Studia Mathematica Bulgarica 8: 85–93, ISSN 0204-9805, MR866648 
• Tits, Jacques (1964), "Algebraic and abstract simple groups", Annals of Mathematics. Second Series 80: 313–329, ISSN 0003-486X, JSTOR 1970394, MR0164968 
• 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, MR735227 

External links

• ATLAS of Group Representations — The Tits Group