Sporadic group

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In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families, plus 26 exceptions which do not follow such a systematic pattern. These are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Sometimes the Tits group is regarded as a sporadic group (because it is not strictly a group of Lie type), in which case there are 27 sporadic groups.

The Monster group is the largest of the sporadic groups and contains all but six of the other sporadic groups as subgroups or subquotients.

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[edit] Names of the sporadic groups

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:

Sporadic Finite Groups Showing (Sporadic) Subgroups
Sporadic Finite Groups Showing (Sporadic) Subgroups

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

The earliest use of the term "sporadic group" may be Burnside (1911, p. 504, note N) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received".

Diagram is based on diagram given in Ronan (2006). The sporadic groups also have a lot of subgroups which are not sporadic but these are not shown on the diagram because they are too numerous.

[edit] Organization

Of the 26 sporadic groups, 20 can be seen inside the Monster group as subgroups or quotients of subgroups. The six exceptions are J1, J3, J4, O'N, Ru and Ly. These six groups are sometimes known as the pariahs.

The remaining twenty groups have been called the Happy Family by Robert Griess, and can be organized in into three generations.

[edit] First generation: the Mathieu groups

Main article: Mathieu groups

The Mathieu groups Mn (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.

[edit] Second generation: the Leech lattice

See also: Leech lattice and Conway groups

The second generation are all subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:

  • Co1 is the quotient of the automorphism group by its center {±1}
  • Co2 is the stabilizer of a type 2 (i.e., length 2) vector
  • Co3 is the stabilizer of a type 3 (i.e., length √6) 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
  • J2 is the group of automorphisms preserving a quaternionic structure (modulo its center).

[edit] Third generation: other subgroups of the Monster

The third generation consists of subgroups which are closely related to the Monster group M:

  • B or F2 has a double cover which is the centralizer of an element of order 2 in M
  • Fi24′ has a triple cover which is the centralizer of an element of order 3 in M (in conjugacy class "3A")
  • Fi23 is a subgroup of Fi24′
  • Fi22 has a double cover which is a subgroup of Fi23
  • The product of Th = F3 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 = F5 and a group of order 5 is the centralizer of an element of order 5 in M
  • The product of He = F7 and a group of order 7 is the centralizer of an element of order 7 in M.

(This series continues further: the product of M12 and a group of order 11 is the centralizer of an element of order 11 in M.)

The Tits group also belongs in this generation: there is a subgroup S4 × 2F4(2)′ normalising a 2C2 subgroup of B, giving rise to a subgroup 2·S4 × 2F4(2)′ normalising a certain Q8 subgroup of the Monster. 2F4(2)′ is also a subgroup of the Fischer groups Fi22, Fi23 and Fi24′, and of the Baby Monster B.

[edit] Table of the sporadic group orders

Group Order 1SF Factorized order
F1 or M 808017424794512875886459904961710757005754368000000000 ≈ 8 × 1053 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
F2 or B 4154781481226426191177580544000000 ≈ 4 × 1033 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47
Fi24' or F3+ 1255205709190661721292800 ≈ 1 × 1024 221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29
Fi23 4089470473293004800 ≈ 4 × 1018 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23
Fi22 64561751654400 ≈ 6 × 1013 217 · 39 · 52 · 7 · 11 · 13
F3 or Th 90745943887872000 ≈ 9 × 1016 215 · 310 · 53 · 72 · 13 · 19 · 31
Ly 51765179004000000 ≈ 5 × 1016 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67
F5 or HN 273030912000000 ≈ 3 × 1014 214 · 36 · 56 · 7 · 11 · 19
Co1 4157776806543360000 ≈ 4 × 1018 221 · 39 · 54 · 72 · 11 · 13 · 23
Co2 42305421312000 ≈ 4 × 1013 218 · 36 · 53 · 7 · 11 · 23
Co3 495766656000 ≈ 5 × 1011 210 · 37 · 53 · 7 · 11 · 23
O'N 460815505920 ≈ 5 × 1011 29 · 34 · 5 · 73 · 11 · 19 · 31
Suz 448345497600 ≈ 4 × 1011 213 · 37 · 52 · 7 · 11 · 13
Ru 145926144000 ≈ 1 × 1011 214 · 33 · 53 · 7 · 13 · 29
He 4030387200 ≈ 4 × 109 210 · 33 · 52 · 73 · 17
McL 898128000 ≈ 9 × 108 27 · 36 · 53 · 7 · 11
HS 44352000 ≈ 4 × 107 29 · 32 · 53 · 7 · 11
J4 86775571046077562880 ≈ 9 × 1019 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43
J3 or HJM 50232960 ≈ 5 × 107 27 · 35 · 5 · 17 · 19
J2 or HJ 604800 ≈ 6 × 105 27 · 33 · 52 · 7
J1 175560 ≈ 2 × 105 23 · 3 · 5 · 7 · 11 · 19
M24 244823040 ≈ 2 × 108 210 · 33 · 5 · 7 · 11 · 23
M23 10200960 ≈ 1 × 107 27 · 32 · 5 · 7 · 11 · 23
M22 443520 ≈ 4 × 105 27 · 32 · 5 · 7 · 11
M12 95040 ≈ 1 × 105 26 · 33 · 5 · 11
M11 7920 ≈ 8 × 103 24 · 32 · 5 · 11

[edit] References

  • Burnside, William (1911), Theory of groups of finite order, pp. 504 (note N), ISBN 0486495752 (2004 reprinting) 
  • Conway, J. H.: A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398-400.
  • Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press, 1985, ISBN 0-19-853199-0
  • Daniel Gorenstein, Richard Lyons, Ronald Solomon The Classification of the Finite Simple Groups (volume 1), AMS, 1994 (volume 2), AMS.
  • Griess, Robert L.: "Twelve Sporadic Groups", Springer-Verlag, 1998.
  • Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 978-0-19-280722-9 

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