Tits group

In group theory, the Tits group 2F4(2)′, named for Jacques 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, who showed that they are simple if n ≥ 1. The first member 2F4(2) of this series is not simple. It was studied by 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.

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 and.

Maximal subgroups
and 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)

5<SUP>2</SUP>:4A<SUB>4</SUB>

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


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

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).