Tits group
In group theory, the Tits group "F"(2)′, 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 "F"(2) were constructed by , who showed that they are simple if "n" ≥ 1. The first member of this series "F"(2) is not simple. It was studied by who showed that it is almost simple, its derived subgroup "F"(2)′ of index 2 being a new simple group, now called the Tits group. The group "F"(2) 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 "F"(2). The Tits group occurs as a maximal subgroup of the Fischer group Fi. The groups "F"(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: L(3):2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations. 2..5.4 Centralizer of an involution. L(25) 2..S A.2 (Two classes , fused by an outer automorphism) 5:4A Presentation. The Tits group can be defined in terms of generators and relations by where ["a", "b"] is the commutator "a""b""ab". It has an outer automorphism obtained by sending ("a", "b") to ("a", "b"("ba")"b"("ba"))