Sporadic 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:

The diagram shows the subquotient 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, 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 F_3|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. 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 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

M_n 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 also belongs in this generation: there is a subgroup S₄ ײF₄(2)′ normalising a 2C² subgroup of B, giving rise to a subgroup 2·S₄ ײF₄(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 the OEIS)	1SF	Factorized order
F1 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
F2 or B	third	4154781481226426191177580544000000	≈ 4×10^33	2^41 · 3^13 · 5^6 · 7^2 · 11 · 13 · 17 · 19 · 23 · 31 · 47
Fi24' or F3+	third	1255205709190661721292800	≈ 1×10^24	2^21 · 3^16 · 5^2 · 7^3 · 11 · 13 · 17 · 23 · 29
Fi23	third	4089470473293004800	≈ 4×10^18	2^18 · 3^13 · 5^2 · 7 · 11 · 13 · 17 · 23
Fi22	third	64561751654400	≈ 6×10^13	2^17 · 3^9 · 5^2 · 7 · 11 · 13
F3 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
F5 or HN	third	273030912000000	≈ 3×10^14	2^14 · 3^6 · 5^6 · 7 · 11 · 19
Co1	second	4157776806543360000	≈ 4×10^18	2^21 · 3^9 · 5^4 · 7^2 · 11 · 13 · 23
Co2	second	42305421312000	≈ 4×10^13	2^18 · 3^6 · 5^3 · 7 · 11 · 23
Co3	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
F7 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
J4	pariah	86775571046077562880	≈ 9×10^19	2^21 · 3^3 · 5 · 7 · 11^3 · 23 · 29 · 31 · 37 · 43
J3 or HJM	pariah	50232960	≈ 5×10^7	2^7 · 3^5 · 5 · 17 · 19
J2 or HJ	second	604800	≈ 6×10^5	2^7 · 3^3 · 5^2 · 7
J1	pariah	175560	≈ 2×10^5	2^3 · 3 · 5 · 7 · 11 · 19
M24	first	244823040	≈ 2×10^8	2^10 · 3^3 · 5 · 7 · 11 · 23
M23	first	10200960	≈ 1×10^7	2^7 · 3^2 · 5 · 7 · 11 · 23
M22	first	443520	≈ 4×10^5	2^7 · 3^2 · 5 · 7 · 11
M12	first	95040	≈ 1×10^5	2^6 · 3^3 · 5 · 11
M11	first	7920	≈ 8×10^3	2^4 · 3^2 · 5 · 11

References

1. 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. 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 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

External links

• Weisstein, Eric W. "Sporadic Group". MathWorld.

• Atlas of Finite Group Representations: Sporadic groups