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In group theory, a discipline within 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 families, plus 26 exceptions that 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. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group,[1] in which case the sporadic groups number 27.

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

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:

Subgroup relations between the sporadic groups
• Mathieu groups M11, M12, M22, M23, M24
• Janko groups J1, J2 or HJ, J3 or HJM, J4
• Conway groups Co1 or F2−, Co2, Co3
• Fischer groups Fi22, Fi23, Fi24′ or F3+
• Higman–Sims group HS
• McLaughlin group McL
• Held group He or F7+ or F7
• Rudvalis group Ru
• Suzuki group Suz or F3−
• O'Nan group O'N
• Harada–Norton group HN or F5+ or F5
• Lyons group Ly
• Thompson group Th or F3|3 or F3
• Baby Monster group B or F2+ or F2
• Fischer–Griess Monster group M or F1

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.

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

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.

Organization

Of the 26 sporadic groups, 20 can be seen inside the Monster group as subgroups or quotients of subgroups (sections).

I. Pariah
Main article: Pariah group

The six exceptions are J1, J3, J4, O'N, Ru and Ly. These six are sometimes known as the pariahs.

II. Happy Family

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

First generation (5 groups): the Mathieu groups
Main article: 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.

Second generation (7 groups): the Leech lattice

All the 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).

Third generation (8 groups): other subgroups of the MonsterThe 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. 2F4(2)′ is also a subgroup of the (pariah) Rudvalis group Ru, and has no involvements in sporadic simple groups except the containments we have already mentioned.

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.
• Finally, the Monster group itself is considered to be in this generation.

(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. 2F4(2)′ is also a subgroup of the (pariah) Rudvalis group Ru, and has no involvements in sporadic simple groups except the containments we have already mentioned.

Table of the sporadic group orders

Group Generation Order (sequence A001228 in OEIS) 1SF Factorized order
F1 or M third 8080174247945128758864599049617107
57005754368000000000
≈ 8×1053 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
F2 or B third 4154781481226426191177580544000000 ≈ 4×1033 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47
Fi24' or F3+ third 1255205709190661721292800 ≈ 1×1024 221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29
Fi23 third 4089470473293004800 ≈ 4×1018 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23
Fi22 third 64561751654400 ≈ 6×1013 217 · 39 · 52 · 7 · 11 · 13
F3 or Th third 90745943887872000 ≈ 9×1016 215 · 310 · 53 · 72 · 13 · 19 · 31
Ly pariah 51765179004000000 ≈ 5×1016 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67
F5 or HN third 273030912000000 ≈ 3×1014 214 · 36 · 56 · 7 · 11 · 19
Co1 second 4157776806543360000 ≈ 4×1018 221 · 39 · 54 · 72 · 11 · 13 · 23
Co2 second 42305421312000 ≈ 4×1013 218 · 36 · 53 · 7 · 11 · 23
Co3 second 495766656000 ≈ 5×1011 210 · 37 · 53 · 7 · 11 · 23
O'N pariah 460815505920 ≈ 5×1011 29 · 34 · 5 · 73 · 11 · 19 · 31
Suz second 448345497600 ≈ 4×1011 213 · 37 · 52 · 7 · 11 · 13
Ru pariah 145926144000 ≈ 1×1011 214 · 33 · 53 · 7 · 13 · 29
F7 or He third 4030387200 ≈ 4×109 210 · 33 · 52 · 73 · 17
McL second 898128000 ≈ 9×108 27 · 36 · 53 · 7 · 11
HS second 44352000 ≈ 4×107 29 · 32 · 53 · 7 · 11
J4 pariah 86775571046077562880 ≈ 9×1019 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43
J3 or HJM pariah 50232960 ≈ 5×107 27 · 35 · 5 · 17 · 19
J2 or HJ second 604800 ≈ 6×105 27 · 33 · 52 · 7
J1 pariah 175560 ≈ 2×105 23 · 3 · 5 · 7 · 11 · 19
M24 first 244823040 ≈ 2×108 210 · 33 · 5 · 7 · 11 · 23
M23 first 10200960 ≈ 1×107 27 · 32 · 5 · 7 · 11 · 23
M22 first 443520 ≈ 4×105 27 · 32 · 5 · 7 · 11
M12 first 95040 ≈ 1×105 26 · 33 · 5 · 11
M11 first 7920 ≈ 8×103 24 · 32 · 5 · 11

References

For example, by John Conway.

Burnside, William (1911), Theory of groups of finite order, p. 504 (note N), ISBN 0-486-49575-2
Conway, J. H. (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups" (PDF), Proc. Nat. Acad. Sci. U.S.A. 61 (2): 398–400, doi:10.1073/pnas.61.2.398, Zbl 0186.32401
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press. ISBN 0-19-853199-0. Zbl 0568.20001.
Gorenstein, D.; Lyons, R.; Solomon, R. (1994), The Classification of the Finite Simple Groups, American Mathematical Society Issues 1, 2, ...
Griess, Robert L. (1998), Twelve Sporadic Groups, Springer-Verlag, ISBN 3540627782, Zbl 0908.20007
Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 978-0-19-280722-9, Zbl 1113.00002