Tits group
Encyclopedia
In mathematics, the Tits group 2F4(2)′ is a finite simple group
of order
17971200 = 211 · 33 · 52 · 13 found by .
The Ree groups 2F4(22n+1) were constructed by , who showed that they are simple if n ≥ 1. The first member of this series 2F4(2) is not simple. It was studied by who showed that its derived subgroup 2F4(2)′ of index 2 was a new simple group. The group 2F4(2) is a group of Lie type and has a BN pair, but the Tits group does not, so is strictly speaking not a group of Lie type
, though it is usually classed with the groups of Lie type in lists of simple groups as it is so close to one.
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 group 2F4(2) 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.
and independently found the 8 classes of maximal subgroups of the Tits group.
The Tits group is one of the simple N-groups
, and was overlooked in John 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 group
s.
The Tits group was characterized in various ways by and .
where [a, b] is the commutator
. It has an outer automorphism obtained by sending (a, b) to (a, bbabababababbababababa).
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
17971200 = 211 · 33 · 52 · 13 found by .
The Ree groups 2F4(22n+1) were constructed by , who showed that they are simple if n ≥ 1. The first member of this series 2F4(2) is not simple. It was studied by who showed that its derived subgroup 2F4(2)′ of index 2 was a new simple group. The group 2F4(2) is a group of Lie type and has a BN pair, but the Tits group does not, so is strictly speaking not a group of Lie type
Group of Lie type
In mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
, though it is usually classed with the groups of Lie type in lists of simple groups as it is so close to one.
Properties
The Schur multiplierSchur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
of the Tits group is trivial and its outer automorphism group
Outer automorphism group
In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...
has order 2, with the full automorphism group being the group 2F4(2).
The group 2F4(2) occurs as a maximal subgroup of the Rudvalis group
Rudvalis group
In the mathematical field of group theory, the Rudvalis group Ru is a sporadic simple group of order-Properties:The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer the Ree group...
, as the point stabilizer of the rank-3 permutation action on 4060 = 1 + 1755 + 2304 points.
and independently found the 8 classes of maximal subgroups of the Tits group.
The Tits group is one of the simple N-groups
N-group (finite group theory)
In mathematical finite group theory, an N-group is a group all of whose local subgroups are solvable groups...
, and was overlooked in John Thompson's
John G. Thompson
John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize....
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 group
Thin finite group
In the mathematical classification of finite groups, a thin group is a finite group such that for any odd prime the Sylow p-subgroups of the 2-local subgroups are cyclic....
s.
The Tits group was characterized in various ways by and .
Presentation
The Tits group can be defined in terms of generators and relations bywhere [a, b] is the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
. It has an outer automorphism obtained by sending (a, b) to (a, bbabababababbababababa).