Sporadic group

Short description: Finite simple group type not classified as Lie, cyclic or alternating

In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.

A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The mentioned 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 26 exceptions are the sporadic groups. The Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type, in which case there would be 27 sporadic groups.

The monster group, or friendly giant, 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 Émile Mathieu in the 1860s and the other twenty-one were found between 1965 (J₁) and 1975 (J₄). 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 26 sporadic groups. A connecting line means the lower group is subquotient of the upper – and no sporadic subquotient in between:
File:EllipseSubqR.svg 1st generation (Mathieu groups)
File:EllipseSubqG.svg 2nd generation (Leech lattice groups)
File:EllipseSubqB.svg 3rd generation (Fischer's 3-transposition groups and other Monster centralizers)
File:EllipseSubqW.svg Pariahs (half-dozen oddments)
Classifications from (Griess, Jr. 1982) and (Conway Curtis)

• 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 (ON)
• 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

Various constructions for these groups were first compiled in (Conway Curtis), including character tables, individual conjugacy classes and lists of maximal subgroup, as well as Schur multipliers and orders of their outer automorphisms. These are also listed online at (Wilson Parker), updated with their group presentations and semi-presentations. The degrees of minimal faithful representation or Brauer characters over fields of characteristic p ≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in (Jansen 2005). A further exception in the classification of finite simple groups is the Tits group T, which is sometimes considered of Lie type or sporadic — 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.

The diagram at right is based on (Ronan 2006). It does not show the numerous non-sporadic simple subquotients of the sporadic groups. The earliest use of the term sporadic group may be (Burnside 1911) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received." (At the time, the other sporadic groups had not been discovered.)

Organization

Happy Family

Of the 26 (or 27) 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

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

See also: Leech lattice and Conway groups

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.)

Pariahs

The six sporadic groups which are not subquotients of the monster group M are J₁, J₃, J₄, O'N, Ru, and Ly, sometimes known as the pariahs.

Tits group

If regarded as a sporadic group, the Tits group ²F₄(2)′ (T for short) would be a third generation group. There is a subgroup S₄ײF₄(2)′ normalizing a 2C² subgroup of the Baby monster B, giving rise to a subgroup 2·S₄ײF₄(2)′ normalizing a certain Q₈ subgroup of the Monster M. ²F₄(2)′ is also a subquotient of the Fischer group Fi₂₂, and thus also of Fi₂₃ and Fi₂₄′, and of the Baby Monster B; involvement of T outside of third-generation sporadic groups, is only seen as being a subquotient of the (pariah) Rudvalis group Ru. T has no part in sporadic simple groups except the ones already mentioned. In terms of classification as a finite simple group, T is the (n=0)–member ²F₄(2)′ of the infinite family of commutator groups ²F₄(2²ⁿ⁺¹)′, thus in a strict sense not sporadic, nor of Lie type (sources have historically regarded the Tits group as belonging in either grouping, or as neither sporadic nor of Lie type, or both altogether). For n > 0 these finite simple groups coincide with the groups of Lie type ²F₄(2²ⁿ⁺¹), also known as Ree groups of type ²F₄.

Assorted table of information

Here are listed the sporadic groups by their discovery date, and discoverer, alongside group generation (per Griess), order, minimal degree of faithful representation (dimensional Brauer character), and presentations via generators. The Tits group is included with a greyer shade of blue to reflect its ambiguous status within the classification scheme as a third-generation group.

Group	Discoverer	Year	Generation	Order	Degree of minimal faithful Brauer character	$(a,b,ab)$ Generators	$⟨⟨a,b\mid o(z)⟩⟩$ Semi-presentation
M or F1	Fischer, Griess	1973	3 3rd	align=right | 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 = 246·320·59·76·112·133·17·19·23·29·31·41·47·59·71 ≈ 8×1053	196883	2A, 3B, 29	o((ab)4(ab2)2) = 50
B or F2	Fischer	1973	3 3rd	align=right | 4,154,781,481,226,426,191,177,580,544,000,000 = 241·313·56·72·11·13·17·19·23·31·47 ≈ 4×1033	4371	2C, 3A, 55	o((ab)2(abab2)2ab2) = 23
Fi24 or F3+	Fischer	1971	3 3rd	align=right | 1,255,205,709,190,661,721,292,800 = 221·316·52·73·11·13·17·23·29 ≈ 1×1024	8671	2A, 3E, 29	o((ab)3) = 33
Fi23	Fischer	1971	3 3rd	align=right | 4,089,470,473,293,004,800 = 218·313·52·7·11·13·17·23 ≈ 4×1018	782	2B, 3D, 28	o(abb(ab)14) = 5
Fi22	Fischer	1971	3 3rd	align=right | 64,561,751,654,400 = 217·39·52·7·11·13 ≈ 6×1013	78	2A, 13, 11	o((ab)2(abab)2ab2) = 12
Th or F3	Thompson	1976	3 3rd	align=right | 90,745,943,887,872,000 = 215·310·53·72·13·19·31 ≈ 9×1016	248	2, 3A, 19	o((ab)3b) = 21
Ly	Lyons	1972	4 Pariah	align=right | 51,765,179,004,000,000 = 28·37·56·7·11·31·37·67 ≈ 5×1016	2480	2, 5A, 14	o(ababab2) = 67
HN or F5	Harada, Norton	1976	3 3rd	align=right | 273,030,912,000,000 = 214·36·56·7·11·19 ≈ 3×1014	133	2A, 3B, 22	o([a, b]) = 5
Co1	Conway	1969	2 2nd	align=right | 4,157,776,806,543,360,000 = 221·39·54·72·11·13·23 ≈ 4×1018	276	2B, 3C, 40	o(ab(abab2)2) = 42
Co2	Conway	1969	2 2nd	align=right | 42,305,421,312,000 = 218·36·53·7·11·23 ≈ 4×1013	23	2A, 5A, 28	o([a, b]) = 4
Co3	Conway	1969	2 2nd	align=right | 495,766,656,000 = 210·37·53·7·11·23 ≈ 5×1011	23	2A, 7C, 17	o((uvv)3(uv)6) = 5
ON or O'N	O'Nan	1976	4 Pariah	align=right | 460,815,505,920 = 29·34·5·73·11·19·31 ≈ 5×1011	10944	2A, 4A, 11	o(abab(b2(b2)abab)5) = 5
Suz	Suzuki	1969	2 2nd	align=right | 448,345,497,600 = 213·37·52·7·11·13 ≈ 4×1011	143	2B, 3B, 13	o([a, b]) = 15
Ru	Rudvalis	1972	4 Pariah	align=right | 145,926,144,000 = 214·33·53·7·13·29 ≈ 1×1011	378	2B, 4A, 13	o(abab2) = 29
He or F7	Held	1969	3 3rd	align=right | 4,030,387,200 = 210·33·52·73·17 ≈ 4×109	51	2A, 7C, 17	o(ab2abab2ab2) = 10
McL	McLaughlin	1969	2 2nd	align=right | 898,128,000 = 27·36·53·7·11 ≈ 9×108	22	2A, 5A, 11	o((ab)2(abab2)2ab2) = 7
HS	Higman, Sims	1967	2 2nd	align=right | 44,352,000 = 29·32·53·7·11 ≈ 4×107	22	2A, 5A, 11	o(abab2) = 15
J4	Janko	1976	4 Pariah	align=right | 86,775,571,046,077,562,880 = 221·33·5·7·113·23·29·31·37·43 ≈ 9×1019	1333	2A, 4A, 37	o(abab2) = 10
J3 or HJM	Janko	1968	4 Pariah	align=right | 50,232,960 = 27·35·5·17·19 ≈ 5×107	85	2A, 3A, 19	o([a, b]) = 9
J2 or HJ	Janko	1968	2 2nd	align=right | 604,800 = 27·33·52·7 ≈ 6×105	14	2B, 3B, 7	o([a, b]) = 12
J1	Janko	1965	4 Pariah	align=right | 175,560 = 23·3·5·7·11·19 ≈ 2×105	56	2, 3, 7	o(abab2) = 19
M24	Mathieu	1861	1 1st	align=right | 244,823,040 = 210·33·5·7·11·23 ≈ 2×108	23	2B, 3A, 23	o(ab(abab2)2ab2) = 4
M23	Mathieu	1861	1 1st	align=right | 10,200,960 = 27·32·5·7·11·23 ≈ 1×107	22	2, 4, 23	o((ab)2(abab2)2ab2) = 8
M22	Mathieu	1861	1 1st	align=right | 443,520 = 27·32·5·7·11 ≈ 4×105	21	2A, 4A, 11	o(abab2) = 11
M12	Mathieu	1861	1 1st	align=right | 95,040 = 26·33·5·11 ≈ 1×105	11	2B, 3B, 11	o([a, b]) = o(ababab2) = 6
M11	Mathieu	1861	1 1st	align=right | 7,920 = 24·32·5·11 ≈ 8×103	10	2, 4, 11	o((ab)2(abab2)2ab2) = 4
T or 2F4(2)′	Tits	1964	3 3rd	align=right | 17,971,200 = 211·33·52·13 ≈ 2×107	104	2A, 3, 13	o([a, b]) = 5

References

Works cited

• Burnside, William (1911). Theory of groups of finite order (2nd ed.). Cambridge: Cambridge University Press. pp. xxiv, 1–512. doi:10.1112/PLMS/S2-7.1.1. ISBN 0-486-49575-2. OCLC 54407807. http://archive.org/details/117768378. 

• 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. PMID 16591697. Bibcode: 1968PNAS...61..398C. 

• 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. Oxford: Clarendon Press. pp. xxxiii; 1–252. ISBN 978-0-19-853199-9. OCLC 12106933. 

• Gorenstein, D. (1979). "The classification of finite simple groups. I. Simple groups and local analysis". Bulletin of the American Mathematical Society. New Series (American Mathematical Society) 1 (1): 43–199. doi:10.1090/S0273-0979-1979-14551-8. 

• Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1998). The classification of the finite simple groups, Number 3. Mathematical Surveys and Monographs. 40. Providence, R.I.: American Mathematical Society. pp. xiii, 1–362. doi:10.1112/S0024609398255457. ISBN 978-0-8218-0391-2. OCLC 6907721813. https://bookstore.ams.org/surv-40-3/. 

• Griess, Jr., Robert L. (1982). "The Friendly Giant". Inventiones Mathematicae 69: 1−102. doi:10.1007/BF01389186. Bibcode: 1982InMat..69....1G. https://www.digizeitschriften.de/dms/img/?PPN=PPN356556735_0069&DMDID=dmdlog7. 

• Griess, Jr., Robert L. (1998). Twelve Sporadic Groups. Springer Monographs in Mathematics. Berlin: Springer-Verlag. pp. 1−169. ISBN 9783540627784. OCLC 38910263. 

• Hartley, Michael I.; Hulpke, Alexander (2010), "Polytopes Derived from Sporadic Simple Groups", Contributions to Discrete Mathematics (Alberta, CA: University of Calgary Department of Mathematics and Statistics) 5 (2): 106−118, doi:10.11575/cdm.v5i2.61945, ISSN 1715-0868, https://cdm.ucalgary.ca/article/view/61945/46662 

• Hiss, Gerhard (2003). "Die Sporadischen Gruppen (The Sporadic Groups)". Jahresber. Deutsch. Math.-Verein. (Annual Report of the German Mathematicians Association) 105 (4): 169−193. ISSN 0012-0456. https://www.mathematik.de/images/DMV/Jahresberichte/Jahresberichte_Archiv/Jahresbericht_04-2003.pdf.  (German)

• Howlett, R. B.; Rylands, L. J.; Taylor, D. E. (2001). "Matrix generators for exceptional groups of Lie type". Journal of Symbolic Computation 31 (4): 429–445. doi:10.1006/jsco.2000.0431. 

• Jansen, Christoph (2005). "The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups". LMS Journal of Computation and Mathematics (London Mathematical Society) 8: 122−144. doi:10.1112/S1461157000000930. 

• Lubeck, Frank (2001). "Smallest degrees of representations of exceptional groups of Lie type". Communications in Algebra (Philadelphia, PA: Taylor & Francis) 29 (5): 2147−2169. doi:10.1081/AGB-100002175. https://www.tandfonline.com/doi/abs/10.1081/AGB-100002175. 

• Nickerson, S.J.; Wilson, R.A. (2011). "Semi-Presentations for the Sporadic Simple Groups". Experimental Mathematics (Oxfordshire: Taylor & Francis) 14 (3): 359−371. doi:10.1080/10586458.2005.10128927. https://www.tandfonline.com/doi/abs/10.1080/10586458.2005.10128927. 

• Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. vii, 1–255. ISBN 978-0-19-280722-9. OCLC 180766312. https://archive.org/details/symmetrymonstero0000rona. 

• Sloane, N. J. A., ed (1996). "Orders of sporadic simple groups (A001228)". OEIS Foundation. http://oeis.org/A001228. 

• Wilson, R.A (1998). "Chapter: An Atlas of Sporadic Group Representations". The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249). Cambridge, U.K: Cambridge University Press. pp. 261–273. doi:10.1017/CBO9780511565830.024. ISBN 9780511565830. OCLC 726827806. https://webspace.maths.qmul.ac.uk/r.a.wilson/pubs_files/ASGRweb.pdf. 

• Wilson, R.A.; Parker, R.A.; Nickerson, S.J.; Bray, J.N. (1999). "ATLAS of Group Representations". Queen Mary University of London. https://brauer.maths.qmul.ac.uk/Atlas/. 

External links

• Weisstein, Eric W.. "Sporadic Group". http://mathworld.wolfram.com/SporadicGroup.html. 
• Atlas of Finite Group Representations: Sporadic groups

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