Sporadic group
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In group theory, 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^{[1]} 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,^{[2]} 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[edit]
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 Co_{1} (Co1), Co_{2} (Co2), Co_{3} (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.^{[3]} In some other sources, the Tits group is regarded as neither sporadic nor of Lie type.^{[4]} Anyway, it is the (n = 0)-member ^{2}F_{4}(2)′ of the infinite family of commutator groups ^{2}F_{4}(2^{2n+1})′ — and thus per definitionem not sporadic. For n > 0 these finite simple groups coincide with the groups of Lie type ^{2}F_{4}(2^{2n+1}). But for n = 0, the derived subgroup ^{2}F_{4}(2)′, called Tits group, is simple and has an index 2 in the finite group ^{2}F_{4}(2) 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 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 at right is based on Ronan (2006). It does not show the numerous non-sporadic simple subquotients of the sporadic groups.
Organization[edit]
Happy family[edit]
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[edit]
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[edit]
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 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[edit]
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 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 subquotient of the Fischer group Fi_{22}, and thus also of Fi_{23} and Fi_{24}′, and of the Baby Monster B. ^{2}F_{4}(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[edit]
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)[edit]
Group | Gen. | Order, OEIS A001228 | Factorized order | Standard generators triple (a, b, ab)^{[5]}^{[6]}^{[3]} |
Further conditions | |
---|---|---|---|---|---|---|
F_{1} or M | 3rd | 80801742479451 |
≈ 8×10^{53} | 2^{46} · 3^{20} · 5^{9} · 7^{6} · 11^{2} · 13^{3} · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 | 2A, 3B, 29 | None |
F_{2} or B | 3rd | 41547814812264 |
≈ 4×10^{33} | 2^{41} · 3^{13} · 5^{6} · 7^{2} · 11 · 13 · 17 · 19 · 23 · 31 · 47 | 2C, 3A, 55 | |
Fi_{24}' or F_{3+} | 3rd | 12552 |
≈ 1×10^{24} | 2^{21} · 3^{16} · 5^{2} · 7^{3} · 11 · 13 · 17 · 23 · 29 | 2A, 3E, 29 | |
Fi_{23} | 3rd | 4089470473293004800 | ≈ 4×10^{18} | 2^{18} · 3^{13} · 5^{2} · 7 · 11 · 13 · 17 · 23 | 2B, 3D, 28 | None |
Fi_{22} | 3rd | 64561751654400 | ≈ 6×10^{13} | 2^{17} · 3^{9} · 5^{2} · 7 · 11 · 13 | 2A, 13, 11 | |
F_{3} or Th | 3rd | 90745943887872000 | ≈ 9×10^{16} | 2^{15} · 3^{10} · 5^{3} · 7^{2} · 13 · 19 · 31 | 2, 3A, 19 | None |
Ly | Pariah | 51765179004000000 | ≈ 5×10^{16} | 2^{8} · 3^{7} · 5^{6} · 7 · 11 · 31 · 37 · 67 | 2, 5A, 14 | |
F_{5} or HN | 3rd | 273030912000000 | ≈ 3×10^{14} | 2^{14} · 3^{6} · 5^{6} · 7 · 11 · 19 | 2A, 3B, 22 | |
Co_{1} | 2nd | 4157776806543360000 | ≈ 4×10^{18} | 2^{21} · 3^{9} · 5^{4} · 7^{2} · 11 · 13 · 23 | 2B, 3C, 40 | None |
Co_{2} | 2nd | 42305421312000 | ≈ 4×10^{13} | 2^{18} · 3^{6} · 5^{3} · 7 · 11 · 23 | 2A, 5A, 28 | None |
Co_{3} | 2nd | 495766656000 | ≈ 5×10^{11} | 2^{10} · 3^{7} · 5^{3} · 7 · 11 · 23 | 2A, 7C, 17 | None |
O'N | Pariah | 460815505920 | ≈ 5×10^{11} | 2^{9} · 3^{4} · 5 · 7^{3} · 11 · 19 · 31 | 2A, 4A, 11 | None |
Suz | 2nd | 448345497600 | ≈ 4×10^{11} | 2^{13} · 3^{7} · 5^{2} · 7 · 11 · 13 | 2B, 3B, 13 | |
Ru | Pariah | 145926144000 | ≈ 1×10^{11} | 2^{14} · 3^{3} · 5^{3} · 7 · 13 · 29 | 2B, 4A, 13 | None |
F_{7} or He | 3rd | 4030387200 | ≈ 4×10^{9} | 2^{10} · 3^{3} · 5^{2} · 7^{3} · 17 | 2A, 7C, 17 | None |
McL | 2nd | 898128000 | ≈ 9×10^{8} | 2^{7} · 3^{6} · 5^{3} · 7 · 11 | 2A, 5A, 11 | |
HS | 2nd | 44352000 | ≈ 4×10^{7} | 2^{9} · 3^{2} · 5^{3} · 7 · 11 | 2A, 5A, 11 | None |
J_{4} | Pariah | 86775571046077562880 | ≈ 9×10^{19} | 2^{21} · 3^{3} · 5 · 7 · 11^{3} · 23 · 29 · 31 · 37 · 43 | 2A, 4A, 37 | |
J_{3} or HJM | Pariah | 50232960 | ≈ 5×10^{7} | 2^{7} · 3^{5} · 5 · 17 · 19 | 2A, 3A, 19 | |
J_{2} or HJ | 2nd | 604800 | ≈ 6×10^{5} | 2^{7} · 3^{3} · 5^{2} · 7 | 2B, 3B, 7 | |
J_{1} | Pariah | 175560 | ≈ 2×10^{5} | 2^{3} · 3 · 5 · 7 · 11 · 19 | 2, 3, 7 | |
T | 3rd | 17971200 | ≈ 2×10^{7} | 2^{11} · 3^{3} · 5^{2} · 13 | 2A, 3, 13 | |
M_{24} | 1st | 244823040 | ≈ 2×10^{8} | 2^{10} · 3^{3} · 5 · 7 · 11 · 23 | 2B, 3A, 23 | |
M_{23} | 1st | 10200960 | ≈ 1×10^{7} | 2^{7} · 3^{2} · 5 · 7 · 11 · 23 | 2, 4, 23 | |
M_{22} | 1st | 443520 | ≈ 4×10^{5} | 2^{7} · 3^{2} · 5 · 7 · 11 | 2A, 4A, 11 | |
M_{12} | 1st | 95040 | ≈ 1×10^{5} | 2^{6} · 3^{3} · 5 · 11 | 2B, 3B, 11 | None |
M_{11} | 1st | 7920 | ≈ 8×10^{3} | 2^{4} · 3^{2} · 5 · 11 | 2, 4, 11 |
References[edit]
- ^ The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups ^{2}F_{4}(2^{2n+1})′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.
- ^ For example, by John Conway.
- ^ ^{a} ^{b} Wilson RA, Parker RA, Nickerson SJ, Bray JN (1999). "Atlas: Sporadic Groups".
- ^ In Eric W. Weisstein „Tits Group“ From MathWorld--A Wolfram Web Resource there is a link from the Tits group to „Sporadic Group“, whereas in Eric W. Weisstein „Sporadic Group“ From MathWorld--A Wolfram Web Resource, however, the Tits group is not listed among the 26. Both sources checked on 2018-05-26.
- ^ Wilson RA (1998). "An Atlas of Sporadic Group Representations" (PDF).
- ^ Nickerson SJ, Wilson RA (2000). "Semi-Presentations for the Sporadic Simple Groups".
- 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", Proc. Natl. Acad. Sci. U.S.A., 61 (2): 398–400, doi:10.1073/pnas.61.2.398, PMC 225171, PMID 16591697, Zbl 0186.32401
- Griess, Robert L. (1982), "The Friendly Giant", Inventiones Mathematicae, 69: 1−102, doi:10.1007/BF01389186, hdl:2027.42/46608, S2CID 123597150
- 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