Sporadic group

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

• 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.[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

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, if regarded as a sporadic group, would belong 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 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	≈ 8×10^53	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
F2 or B	3rd	4154781481226426191177580544000000	≈ 4×10^33	2^41 · 3^13 · 5^6 · 7^2 · 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	≈ 1×10^24	2^21 · 3^16 · 5^2 · 7^3 · 11 · 13 · 17 · 23 · 29	2A, 3E, 29	${\displaystyle o\left((ab)^{3}b\right)=33}$
Fi23	3rd	4089470473293004800	≈ 4×10^18	2^18 · 3^13 · 5^2 · 7 · 11 · 13 · 17 · 23	2B, 3D, 28	None
Fi22	3rd	64561751654400	≈ 6×10^13	2^17 · 3^9 · 5^2 · 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	≈ 9×10^16	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	${\displaystyle o\left(ababab^{2}\right)=67}$
F5 or HN	3rd	273030912000000	≈ 3×10^14	2^14 · 3^6 · 5^6 · 7 · 11 · 19	2A, 3B, 22	${\displaystyle o([a,b])=5}$
Co1	2nd	4157776806543360000	≈ 4×10^18	2^21 · 3^9 · 5^4 · 7^2 · 11 · 13 · 23	2B, 3C, 40	None
Co2	2nd	42305421312000	≈ 4×10^13	2^18 · 3^6 · 5^3 · 7 · 11 · 23	2A, 5A, 28	None
Co3	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	${\displaystyle o([a,b])=15}$
Ru	Pariah	145926144000	≈ 1×10^11	2^14 · 3^3 · 5^3 · 7 · 13 · 29	2B, 4A, 13	None
F7 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	${\displaystyle o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=7}$
HS	2nd	44352000	≈ 4×10^7	2^9 · 3^2 · 5^3 · 7 · 11	2A, 5A, 11	None
J4	Pariah	86775571046077562880	≈ 9×10^19	2^21 · 3^3 · 5 · 7 · 11^3 · 23 · 29 · 31 · 37 · 43	2A, 4A, 37	${\displaystyle o\left(abab^{2}\right)=10}$
J3 or HJM	Pariah	50232960	≈ 5×10^7	2^7 · 3^5 · 5 · 17 · 19	2A, 3A, 19	${\displaystyle o([a,b])=9}$
J2 or HJ	2nd	604800	≈ 6×10^5	2^7 · 3^3 · 5^2 · 7	2B, 3B, 7	${\displaystyle o([a,b])=12}$
J1	Pariah	175560	≈ 2×10^5	2^3 · 3 · 5 · 7 · 11 · 19	2, 3, 7	${\displaystyle o\left(abab^{2}\right)=19}$
T	3rd	17971200	≈ 2×10^7	2^11 · 3^3 · 5^2 · 13	2A, 3, 13	${\displaystyle o([a,b])=5}$
M24	1st	244823040	≈ 2×10^8	2^10 · 3^3 · 5 · 7 · 11 · 23	2B, 3A, 23	${\displaystyle o\left(ab\left(abab^{2}\right)^{2}ab^{2}\right)=4}$
M23	1st	10200960	≈ 1×10^7	2^7 · 3^2 · 5 · 7 · 11 · 23	2, 4, 23	${\displaystyle o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=8}$
M22	1st	443520	≈ 4×10^5	2^7 · 3^2 · 5 · 7 · 11	2A, 4A, 11	${\displaystyle o\left(abab^{2}\right)=11}$
M12	1st	95040	≈ 1×10^5	2^6 · 3^3 · 5 · 11	2B, 3B, 11	None
M11	1st	7920	≈ 8×10^3	2^4 · 3^2 · 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. 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".

• 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, p. 1−102, doi:10.1007/BF01389186, hdl:2027.42/46608
• 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