Tits group


In group theory,
the Tits group 2F4′, named for Jacques Tits, is a finite simple group of order
It is sometimes considered a 27th sporadic group.

History and properties

The Ree groups 2F4 were constructed by, who showed that they are simple if n ≥ 1. The first member of this series 2F4 is not simple. It was studied by who showed that it is almost simple, its derived subgroup 2F4′ of index 2 being a new simple group, now called the Tits group. The group 2F4 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 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.
The Tits group occurs as a maximal subgroup of the Fischer group Fi22. The groups 2F4 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:
L3:2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations.
2..5.4 Centralizer of an involution.
L2
22..S3
A6.22
52:4A4

Presentation

The Tits group can be defined in terms of generators and relations by
where is the commutator a−1b−1ab. It has an outer automorphism obtained by sending to 5b