Sporadic group
Algebraic structure → Group theory Group theory 


Modular groups

Infinite dimensional Lie group

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.
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:
 Mathieu groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}
 Janko groups J_{1}, J_{2} or HJ, J_{3} or HJM, J_{4}
 Conway groups Co_{1}, Co_{2}, Co_{3}
 Fischer groups Fi_{22}, Fi_{23}, Fi_{24}′ or F_{3+}
 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
 Harada–Norton group HN or F_{5+} or F_{5}
 Lyons group Ly
 Thompson group Th or F_{33} 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.
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".
The diagram on the right is based on the 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
The six exceptions are J_{1}, J_{3}, J_{4}, 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
M_{n} for n = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M_{24}, 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 {±1}
 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 √6) vector
 Suz is the group of automorphisms preserving a complex structure (modulo its center)
 McL is the stabilizer of a type 223 triangle
 HS is the stabilizer of a type 233 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 also belongs in this generation: there is a subgroup S_{4} ×^{2}F_{4}(2)′ normalising a 2C^{2} subgroup of B, giving rise to a subgroup 2·S_{4} ×^{2}F_{4}(2)′ normalising a certain Q_{8} subgroup of the Monster. ^{2}F_{4}(2)′ is also a subgroup of the Fischer groups Fi_{22}, Fi_{23} and Fi_{24}′, and of the Baby Monster B. ^{2}F_{4}(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 the OEIS)  1SF  Factorized order 

F_{1} or M  third  8080174247945128758864599049617107 57005754368000000000  ≈ 8×10^{53}  2^{46} · 3^{20} · 5^{9} · 7^{6} · 11^{2} · 13^{3} · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 
F_{2} or B  third  4154781481226426191177580544000000  ≈ 4×10^{33}  2^{41} · 3^{13} · 5^{6} · 7^{2} · 11 · 13 · 17 · 19 · 23 · 31 · 47 
Fi_{24}' or F_{3+}  third  1255205709190661721292800  ≈ 1×10^{24}  2^{21} · 3^{16} · 5^{2} · 7^{3} · 11 · 13 · 17 · 23 · 29 
Fi_{23}  third  4089470473293004800  ≈ 4×10^{18}  2^{18} · 3^{13} · 5^{2} · 7 · 11 · 13 · 17 · 23 
Fi_{22}  third  64561751654400  ≈ 6×10^{13}  2^{17} · 3^{9} · 5^{2} · 7 · 11 · 13 
F_{3} or Th  third  90745943887872000  ≈ 9×10^{16}  2^{15} · 3^{10} · 5^{3} · 7^{2} · 13 · 19 · 31 
Ly  pariah  51765179004000000  ≈ 5×10^{16}  2^{8} · 3^{7} · 5^{6} · 7 · 11 · 31 · 37 · 67 
F_{5} or HN  third  273030912000000  ≈ 3×10^{14}  2^{14} · 3^{6} · 5^{6} · 7 · 11 · 19 
Co_{1}  second  4157776806543360000  ≈ 4×10^{18}  2^{21} · 3^{9} · 5^{4} · 7^{2} · 11 · 13 · 23 
Co_{2}  second  42305421312000  ≈ 4×10^{13}  2^{18} · 3^{6} · 5^{3} · 7 · 11 · 23 
Co_{3}  second  495766656000  ≈ 5×10^{11}  2^{10} · 3^{7} · 5^{3} · 7 · 11 · 23 
O'N  pariah  460815505920  ≈ 5×10^{11}  2^{9} · 3^{4} · 5 · 7^{3} · 11 · 19 · 31 
Suz  second  448345497600  ≈ 4×10^{11}  2^{13} · 3^{7} · 5^{2} · 7 · 11 · 13 
Ru  pariah  145926144000  ≈ 1×10^{11}  2^{14} · 3^{3} · 5^{3} · 7 · 13 · 29 
F_{7} or He  third  4030387200  ≈ 4×10^{9}  2^{10} · 3^{3} · 5^{2} · 7^{3} · 17 
McL  second  898128000  ≈ 9×10^{8}  2^{7} · 3^{6} · 5^{3} · 7 · 11 
HS  second  44352000  ≈ 4×10^{7}  2^{9} · 3^{2} · 5^{3} · 7 · 11 
J_{4}  pariah  86775571046077562880  ≈ 9×10^{19}  2^{21} · 3^{3} · 5 · 7 · 11^{3} · 23 · 29 · 31 · 37 · 43 
J_{3} or HJM  pariah  50232960  ≈ 5×10^{7}  2^{7} · 3^{5} · 5 · 17 · 19 
J_{2} or HJ  second  604800  ≈ 6×10^{5}  2^{7} · 3^{3} · 5^{2} · 7 
J_{1}  pariah  175560  ≈ 2×10^{5}  2^{3} · 3 · 5 · 7 · 11 · 19 
M_{24}  first  244823040  ≈ 2×10^{8}  2^{10} · 3^{3} · 5 · 7 · 11 · 23 
M_{23}  first  10200960  ≈ 1×10^{7}  2^{7} · 3^{2} · 5 · 7 · 11 · 23 
M_{22}  first  443520  ≈ 4×10^{5}  2^{7} · 3^{2} · 5 · 7 · 11 
M_{12}  first  95040  ≈ 1×10^{5}  2^{6} · 3^{3} · 5 · 11 
M_{11}  first  7920  ≈ 8×10^{3}  2^{4} · 3^{2} · 5 · 11 
References
 ↑ For example, by John Conway.
 Burnside, William (1911), Theory of groups of finite order, p. 504 (note N), ISBN 0486495752
 Conway, J. H. (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups" (PDF), Proc. Natl. 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 0198531990. 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, SpringerVerlag, ISBN 3540627782, Zbl 0908.20007
 Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 9780192807229, Zbl 1113.00002