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In group theory, the **Tits group**^{2}*F*_{4}(2)′, named for Jacques Tits (French: [tits] ), is a finite simple group of order

- 2
^{11}· 3^{3}· 5^{2}· 13 = 17,971,200.

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.

The Ree groups ^{2}*F*_{4}(2^{2n+1}) were constructed by Ree (1961), 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 JacquesTits ( 1964 ) 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^{2n+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.^{ [1] }

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 Fi_{22}. 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 Parrott ( 1972 , 1973 ) and Stroth (1980).

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

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

2.[2^{8}].5.4 Centralizer of an involution.

L_{2}(25)

2^{2}.[2^{8}].S_{3}

A_{6}.2^{2} (Two classes, fused by an outer automorphism)

5^{2}:4A_{4}

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

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}).

- ↑ For instance, by the ATLAS of Finite Groups and its web-based descendant

In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the **sporadic simple groups**, or the **sporadic finite groups**, or just the **sporadic groups**.

In the area of modern algebra known as group theory, the **Suzuki group***Suz* or *Sz* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Thompson group***Th* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Held group***He* is a sporadic simple group of order

In the area of abstract algebra known as group theory, the **O'Nan group***O'N* or **O'Nan–Sims group** is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Rudvalis group***Ru* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Harada–Norton group***HN* is a sporadic simple group of order

In mathematics, specifically in group theory, the phrase * group of Lie type* usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase

In mathematics, a **Ree group** is a group of Lie type over a finite field constructed by Ree from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.

In the area of modern algebra known as group theory, the **Janko group***J _{1}* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Janko group***J _{4}* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Mathieu group***M _{12}* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **McLaughlin group** McL is a sporadic simple group of order

In the mathematical classification of finite simple groups, a **thin group** is a finite group such that for every odd prime number *p*, the Sylow *p*-subgroups of the 2-local subgroups are cyclic. Informally, these are the groups that resemble rank 1 groups of Lie type over a finite field of characteristic 2.

In the area of modern algebra known as group theory, the **Suzuki groups**, denoted by Sz(2^{2n+1}), ^{2}*B*_{2}(2^{2n+1}), Suz(2^{2n+1}), or *G*(2^{2n+1}), form an infinite family of groups of Lie type found by Suzuki, that are simple for *n* ≥ 1. These simple groups are the only finite non-abelian ones with orders not divisible by 3.

In mathematical finite group theory, the **Dempwolff group** is a finite group of order 319979520 = 2^{15}·3^{2}·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group as the subgroup fixing a certain lattice in the Lie algebra of , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

In mathematics, a **Sastry automorphism**, is an automorphism of a field of characteristic 2 satisfying some rather complicated conditions related to the problem of embedding Ree groups of type ^{2}F_{4} into Chevalley groups of type F_{4}. They were introduced by Sastry (1995), and named and classified by Bombieri (2002) who showed that there are 22 families of Sastry automorphisms, together with 22 exceptional ones over some finite fields of orders up to 2^{10}.

In the area of modern algebra known as group theory, the **Fischer group***Fi _{22}* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Conway group***Co _{1}* is a sporadic simple group of order

- Parrott, David (1972), "A characterization of the Tits' simple group",
*Canadian Journal of Mathematics*,**24**(4): 672–685, doi: 10.4153/cjm-1972-063-0 , ISSN 0008-414X, MR 0325757 - Parrott, David (1973), "A characterization of the Ree groups
^{2}*F*_{4}(q)",*Journal of Algebra*,**27**(2): 341–357, doi: 10.1016/0021-8693(73)90109-9 , ISSN 0021-8693, MR 0347965 - Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F
_{4})",*Bulletin of the American Mathematical Society*,**67**: 115–116, doi: 10.1090/S0002-9904-1961-10527-2 , ISSN 0002-9904, MR 0125155 - Stroth, Gernot (1980), "A general characterization of the Tits simple group",
*Journal of Algebra*,**64**(1): 140–147, doi: 10.1016/0021-8693(80)90138-6 , ISSN 0021-8693, MR 0575787 - Tchakerian, Kerope B. (1986), "The maximal subgroups of the Tits simple group",
*Pliska Studia Mathematica Bulgarica*,**8**: 85–93, ISSN 0204-9805, MR 0866648 - Tits, Jacques (1964), "Algebraic and abstract simple groups",
*Annals of Mathematics*, Second Series,**80**(2): 313–329, doi:10.2307/1970394, ISSN 0003-486X, JSTOR 1970394, MR 0164968 - Wilson, Robert A. (1984), "The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits",
*Proceedings of the London Mathematical Society*, Third Series,**48**(3): 533–563, doi:10.1112/plms/s3-48.3.533, ISSN 0024-6115, MR 0735227

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