Sporadic group

In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.

A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups.

The monster group is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it.

Names

Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is:

* Mathieu groups ''M''₁₁ (M11), ''M''₁₂ (M12), ''M''₂₂ (M22), ''M''₂₃ (M23), ''M''₂₄ (M24)
* Janko groups ''J''₁ (J1), ''J''₂ or ''HJ'' (J2), ''J''₃ or ''HJM'' (J3), ''J''₄ (J4)
* Conway groups '' Co₁'' (Co1), '' Co₂'' (), '' Co₃'' (Co3)
* Fischer groups ''Fi''₂₂ (Fi22), ''Fi''₂₃ (Fi23), ''Fi''₂₄′ or ''F''₃₊ (Fi24)
* Higman–Sims group ''HS''
* McLaughlin group ''McL''
* Held group ''He'' or ''F''₇₊ or ''F''₇
* Rudvalis group ''Ru''
* Suzuki group ''Suz'' or ''F''₃₋
* O'Nan group ''O'N'' (ON)
* Harada–Norton group ''HN'' or ''F''₅₊ or ''F''₅
* Lyons group ''Ly''
* Thompson group ''Th'' or ''F''_{3, 3} or ''F''₃
* Baby Monster group ''B'' or ''F''₂₊ or ''F''₂
* Fischer–Griess Monster group ''M'' or ''F''₁

The Tits group ''T'' is sometimes also regarded as a sporadic group (it is almost but not strictly a group of Lie type), which is why in some sources the number of sporadic groups is given as 27 instead of 26. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type.I
Eric W. Weisstein „Tits Group“ From MathWorld--A Wolfram Web Resource
there is a link from the Tits group to „Sporadic Group“, whereas i

however, the Tits group is ''not'' listed among the 26. Both sources checked on 2018-05-26. Anyway, it is the of the ''infinite'' family of commutator groups — and thus by definition not sporadic. For these finite simple groups coincide with the groups of Lie type But for the derived subgroup , called Tits group, is simple and has an index 2 in the finite group of Lie type which —as the only one of the whole family— is not simple.

Matrix representations over finite fields for all the sporadic groups have been constructed.

The earliest use of the term ''sporadic group'' may be where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received."

The diagram at right is based on . It does not show the numerous non-sporadic simple subquotients of the sporadic groups.

Organization

Happy family

Of the 26 sporadic groups, 20 can be seen inside the monster group as subgroups or quotients of subgroups (sections).
These twenty have been called the ''happy family'' by Robert Griess, and can be organized into three generations.

First generation (5 groups): the Mathieu groups

M_''n'' for ''n'' = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on ''n'' points. They are all subgroups of M₂₄, which is a permutation group on 24 points.

Second generation (7 groups): the Leech lattice

All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:
* ''Co''₁ is the quotient of the automorphism group by its center
* ''Co''₂ is the stabilizer of a type 2 (i.e., length 2) vector
* ''Co''₃ is the stabilizer of a type 3 (i.e., length ) vector
* ''Suz'' is the group of automorphisms preserving a complex structure (modulo its center)
* ''McL'' is the stabilizer of a type 2-2-3 triangle
* ''HS'' is the stabilizer of a type 2-3-3 triangle
* ''J''₂ is the group of automorphisms preserving a quaternionic structure (modulo its center).

Third generation (8 groups): other subgroups of the Monster

Consists of subgroups which are closely related to the Monster group ''M'':
* ''B'' or ''F''₂ has a double cover which is the centralizer of an element of order 2 in ''M''
* ''Fi''₂₄′ has a triple cover which is the centralizer of an element of order 3 in ''M'' (in conjugacy class "3A")
* ''Fi''₂₃ is a subgroup of ''Fi''₂₄′
* ''Fi''₂₂ has a double cover which is a subgroup of ''Fi''₂₃
* The product of ''Th'' = ''F''₃ and a group of order 3 is the centralizer of an element of order 3 in ''M'' (in conjugacy class "3C")
* The product of ''HN'' = ''F''₅ and a group of order 5 is the centralizer of an element of order 5 in ''M''
* The product of ''He'' = ''F''₇ and a group of order 7 is the centralizer of an element of order 7 in ''M''.
* Finally, the Monster group itself is considered to be in this generation.

(This series continues further: the product of ''M''₁₂ and a group of order 11 is the centralizer of an element of order 11 in ''M''.)

The Tits group, if regarded as a sporadic group, would belong in this generation: there is a subgroup S₄ ײF₄(2)′ normalising a 2C² subgroup of ''B'', giving rise to a subgroup 2·S₄ ײF₄(2)′ normalising a certain Q₈ subgroup of the Monster. ²F₄(2)′ is also a subquotient of the Fischer group ''Fi''₂₂, and thus also of ''Fi''₂₃ and ''Fi''₂₄′, and of the Baby Monster ''B''. ²F₄(2)′ is also a subquotient of the (pariah) Rudvalis group ''Ru'', and has no involvements in sporadic simple groups except the ones already mentioned.

Pariahs

The six exceptions are ''J''₁, ''J''₃, ''J''₄, ''O'N'', ''Ru'' and ''Ly'', sometimes known as the pariahs.

Table of the sporadic group orders (w/ Tits group)

References

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External links

* {{MathWorld, urlname=SporadicGroup, title=Sporadic Group

Atlas of Finite Group Representations: Sporadic groups

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Mathematical tables

he:משפט המיון לחבורות פשוטות סופיות