Tits group

In group theory,

the Tits group ²F₄(2)′, named for Jacques Tits (), is a finite simple group of order

:   17,971,200 = 2¹¹ · 3³ · 5² · 13.

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.

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 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 .

Maximal subgroups

and independently found the 8 classes of maximal subgroups of the Tits group as follows:

{| class="wikitable"

|+ Maximal subgroups of ²F₄(2)′

|-

! No. !! Structure !! Order !! Index !! Comments

|-

|1,2||L₃(3):2 ||align=right|11,232<br />=&nbsp;2⁵·3³·13 ||align=right| 1,600<br />=&nbsp;2⁶·5² ||two classes, fused by an outer automorphism; fixes a point in a rank 4 permutation representation

|-

| 3||2.[2⁸]:5:4 ||align=right|10,240<br />=&nbsp;2¹¹·5 ||align=right| 1,755<br />=&nbsp;3³·5·13 ||centralizer of an involution of class 2A

|-

| 4||L₂(25) ||align=right| 7,800<br />=&nbsp;2³·3·5²·13||align=right| 2,304<br />=&nbsp;2⁸·3² ||

|-

| 5||2².[2⁸]:S₃||align=right| 6,144<br />=&nbsp;2¹¹·3 ||align=right| 2,925<br />=&nbsp;3²·5²·13||

|-

|6,7||A₆^·&thinsp;2² ||align=right| 1,440<br />=&nbsp;2⁵·3²·5 ||align=right|12,480<br />=&nbsp;2⁶·3·5·13 ||two classes, fused by an outer automorphism

|-

| 8||5²:4A₄ ||align=right| 1,200<br />=&nbsp;2⁴·3·5² ||align=right|14,976<br />=&nbsp;2⁷·3²·13||

|}

Presentation

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

:<math>a^2 = b^3 = (ab)^{13} = [a, b]^5 = [a, bab]^4 = ((ab)^4 ab^{-1})^6 = 1, \,</math>

where [a,&nbsp;b] is the commutator a⁻¹b⁻¹ab. It has an outer automorphism obtained by sending (a,&nbsp;b) to (a,&nbsp;b(ba)⁵b(ba)⁵).

Notes