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

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:

The diagram shows the subquotient relations between the sporadic groups. A connecting line means the lower group is a subquotient of the upper, with no sporadic subquotient in between.
1st generation, 2nd generation, 3rd generation, Pariah

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 2F4(2)′ of the infinite family of commutator groups 2F4(22n+1)′ — and thus per definitionem not sporadic. For n > 0 these finite simple groups coincide with the groups of Lie type 2F4(22n+1). But for n = 0, the derived subgroup 2F4(2)′, called Tits group, is simple and has an index 2 in the finite group 2F4(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

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

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

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 Monster

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, if regarded as a sporadic group, would belong 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 subquotient of the Fischer group Fi22, and thus also of Fi23 and Fi24′, 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 J1, J3, J4, O'N, Ru and Ly, sometimes known as the pariahs.

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

Group Gen. Order, OEIS A001228 Factorized order Standard generators
triple (a, b, ab)[5][6][3]
Further conditions
F1 or M 3rd 808017424794512875886459904961710757005754368000000000 ≈ 8×1053 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 2A, 3B, 29 None
F2 or B 3rd 4154781481226426191177580544000000 ≈ 4×1033 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47 2C, 3A, 55 ${\displaystyle o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=23}$
Fi24' or F3+ 3rd 1255205709190661721292800 ≈ 1×1024 221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29 2A, 3E, 29 ${\displaystyle o\left((ab)^{3}b\right)=33}$
Fi23 3rd 4089470473293004800 ≈ 4×1018 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23 2B, 3D, 28 None
Fi22 3rd 64561751654400 ≈ 6×1013 217 · 39 · 52 · 7 · 11 · 13 2A, 13, 11 ${\displaystyle o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=12}$
F3 or Th 3rd 90745943887872000 ≈ 9×1016 215 · 310 · 53 · 72 · 13 · 19 · 31 2, 3A, 19 None
Ly Pariah 51765179004000000 ≈ 5×1016 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67 2, 5A, 14 ${\displaystyle o\left(ababab^{2}\right)=67}$
F5 or HN 3rd 273030912000000 ≈ 3×1014 214 · 36 · 56 · 7 · 11 · 19 2A, 3B, 22 ${\displaystyle o([a,b])=5}$
Co1 2nd 4157776806543360000 ≈ 4×1018 221 · 39 · 54 · 72 · 11 · 13 · 23 2B, 3C, 40 None
Co2 2nd 42305421312000 ≈ 4×1013 218 · 36 · 53 · 7 · 11 · 23 2A, 5A, 28 None
Co3 2nd 495766656000 ≈ 5×1011 210 · 37 · 53 · 7 · 11 · 23 2A, 7C, 17 None
O'N Pariah 460815505920 ≈ 5×1011 29 · 34 · 5 · 73 · 11 · 19 · 31 2A, 4A, 11 None
Suz 2nd 448345497600 ≈ 4×1011 213 · 37 · 52 · 7 · 11 · 13 2B, 3B, 13 ${\displaystyle o([a,b])=15}$
Ru Pariah 145926144000 ≈ 1×1011 214 · 33 · 53 · 7 · 13 · 29 2B, 4A, 13 None
F7 or He 3rd 4030387200 ≈ 4×109 210 · 33 · 52 · 73 · 17 2A, 7C, 17 None
McL 2nd 898128000 ≈ 9×108 27 · 36 · 53 · 7 · 11 2A, 5A, 11 ${\displaystyle o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=7}$
HS 2nd 44352000 ≈ 4×107 29 · 32 · 53 · 7 · 11 2A, 5A, 11 None
J4 Pariah 86775571046077562880 ≈ 9×1019 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 2A, 4A, 37 ${\displaystyle o\left(abab^{2}\right)=10}$
J3 or HJM Pariah 50232960 ≈ 5×107 27 · 35 · 5 · 17 · 19 2A, 3A, 19 ${\displaystyle o([a,b])=9}$
J2 or HJ 2nd 604800 ≈ 6×105 27 · 33 · 52 · 7 2B, 3B, 7 ${\displaystyle o([a,b])=12}$
J1 Pariah 175560 ≈ 2×105 23 · 3 · 5 · 7 · 11 · 19 2, 3, 7 ${\displaystyle o\left(abab^{2}\right)=19}$
T 3rd 17971200 ≈ 2×107 211 · 33 · 52 · 13 2A, 3, 13 ${\displaystyle o([a,b])=5}$
M24 1st 244823040 ≈ 2×108 210 · 33 · 5 · 7 · 11 · 23 2B, 3A, 23 ${\displaystyle o\left(ab\left(abab^{2}\right)^{2}ab^{2}\right)=4}$
M23 1st 10200960 ≈ 1×107 27 · 32 · 5 · 7 · 11 · 23 2, 4, 23 ${\displaystyle o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=8}$
M22 1st 443520 ≈ 4×105 27 · 32 · 5 · 7 · 11 2A, 4A, 11 ${\displaystyle o\left(abab^{2}\right)=11}$
M12 1st 95040 ≈ 1×105 26 · 33 · 5 · 11 2B, 3B, 11 None
M11 1st 7920 ≈ 8×103 24 · 32 · 5 · 11 2, 4, 11 ${\displaystyle o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=4}$

References

1. ^ The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups 2F4(22n+1)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.
2. ^ For example, by John Conway.
3. ^ a b Wilson RA, Parker RA, Nickerson SJ, Bray JN (1999). "Atlas: Sporadic Groups".
4. ^ 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.
5. ^ Wilson RA (1998). "An Atlas of Sporadic Group Representations" (PDF).
6. ^ Nickerson SJ, Wilson RA (2000). "Semi-Presentations for the Sporadic Simple Groups".