In group theory, the Tits group <sup>2</sup>F<sub>4</sub>(2)â², named for Jacques Tits (), is a finite simple group of order
This is the only simple group that is a derivative of a group of Lie type that is not a group of Lie type in any series from exceptional isomorphisms. It is sometimes considered a 27th sporadic group.
The Ree groups <sup>2</sup>F<sub>4</sub>(2<sup>2n+1</sup>) were constructed by , who showed that they are simple if n âÂÂ¥ 1. The first member <sup>2</sup>F<sub>4</sub>(2) of this series is not simple. It was studied by who showed that it is almost simple, its derived subgroup <sup>2</sup>F<sub>4</sub>(2)â² of index 2 being a new simple group, now called the Tits group. The group <sup>2</sup>F<sub>4</sub>(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 <sup>2</sup>F<sub>4</sub>(2<sup>2n+1</sup>)â² 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 <sup>2</sup>F<sub>4</sub>(2).
The Tits group occurs as a maximal subgroup of the Fischer group Fi<sub>22</sub>. The group <sup>2</sup>F<sub>4</sub>(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 not included 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 .
and independently found the 8 classes of maximal subgroups of the Tits group as follows:
The Tits group can be defined in terms of generators and relations by
where [a, b] is the commutator a<sup>âÂÂ1</sup>b<sup>âÂÂ1</sup>ab. It has an outer automorphism obtained by sending (a, b) to (a, b(ba)<sup>5</sup>b(ba)<sup>5</sup>).