Tits group

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

In group theory, the Tits group ²F₄(2)′, named for Jacques Tits (French: [tits]), is a finite simple group of order

   2¹¹ · 3³ · 5² · 13 = 17,971,200.

This is the only simple group that is a derivative of a group of Lie type that is not strictly a group of Lie type in any series due to exceptional isomorphism. It is sometimes considered a 27th sporadic group.

History and properties

The

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

The Tits group is one of the

The Tits group was characterized in various ways by Parrott (1972, 1973) and Stroth (1980).

Maximal subgroups

Wilson (1984) and Tchakerian (1986) independently found the 8 classes of maximal subgroups of the Tits group as follows:

L₃(3):2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations.

2.[2⁸].5.4 Centralizer of an involution.

L₂(25)

2².[2⁸].S₃

A₆.2² (Two classes, fused by an outer automorphism)

5²:4A₄

Presentation

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

${\displaystyle a^{2}=b^{3}=(ab)^{13}=[a,b]^{5}=[a,bab]^{4}=((ab)^{4}ab^{-1})^{6}=1,\,}$

where [a, b] is the

Notes

References

• Parrott, David (1972), "A characterization of the Tits' simple group",
  MR 0325757
• Parrott, David (1973), "A characterization of the Ree groups ²F₄(q)",
  MR 0347965
• Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F₄)",
  MR 0125155
• Stroth, Gernot (1980), "A general characterization of the Tits simple group",
  MR 0575787
• Tchakerian, Kerope B. (1986), "The maximal subgroups of the Tits simple group",
  MR 0866648
• Tits, Jacques (1964), "Algebraic and abstract simple groups",
  MR 0164968
• MR 0735227