Tits Group

In group theory,
the Tits group ²''F''₄(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 ²''F''₄(2^2''n''+1) were constructed by , who showed that they are simple if ''n'' ≥ 1. The first member ²''F''₄(2) of this series is not simple. It was studied by who showed that it is almost simple, its derived subgroup ²''F''₄(2)′ of index 2 being a new simple group, now called the Tits group. The group ²''F''₄(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 ²''F''₄(2^2''n''+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.For instance, by the ATLAS of Finite Groups and it
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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 ²''F''₄(2).

The Tits group occurs as a maximal subgroup of the Fischer group Fi₂₂. The group ²''F''₄(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:

Presentation

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

:$a^2 = b^3 = (ab)^ =$, b5 = , bab4 = ((ab)^4 ab^)^6 = 1, \,

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

Notes

References

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*{{Citation , last1=Wilson , first1=Robert A. , author-link1=Robert A. Wilson (mathematician) , title=The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits , doi=10.1112/plms/s3-48.3.533 , mr=735227 , year=1984 , journal=Proceedings of the London Mathematical Society , series=Third Series , issn=0024-6115 , volume=48 , issue=3 , pages=533–563

External links

ATLAS of Group Representations — The Tits Group

Sporadic groups

de:Gruppe vom Lie-Typ#Die Tits-Gruppe