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

Asimple group is a groupG that does not have anynormal subgroups except for the trivial group andG itself. The mentioned classification theorem states that thelist of finite simple groups consists of 18countably infinite families^[a] plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. TheTits group is sometimes regarded as a sporadic group because it is not strictly agroup of Lie type,^[1] in which case there would be 27 sporadic groups.

Themonster group, orfriendly giant, is the largest of the sporadic groups, and all but six of the other sporadic groups aresubquotients of it.^[2]

Names

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Five of the sporadic groups were discovered byÉmile 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:^[1]^[3]^[4]

The diagram shows thesubquotient relations between the 26sporadic groups. A connecting line means the lower group is subquotient of the upper – and no sporadic subquotient in between.
The generations of Robert Griess: 1st, 2nd, 3rd, Pariah

Various constructions for these groups were first compiled inConway et al. (1985), includingcharacter tables, individualconjugacy classes and lists ofmaximal subgroup, as well asSchur multipliers and orders of theirouter automorphisms. These are also listed online atWilson et al. (1999), updated with their grouppresentations and semi-presentations. The degrees of minimal faithful representation orBrauer characters over fields of characteristicp ≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed inJansen (2005).

A furtherexception in the classification offinite simple groups is theTits groupT, which is sometimes considered ofLie type^[5] or sporadic — it is almost but not strictly a group of Lie type^[6] — which is why in some sources the number of sporadic groups is given as 27, instead of 26.^[7]^[8] In some other sources, the Tits group is regarded as neither sporadic nor of Lie type, or both.^[9]^{[citation needed]} The Tits group is the(n = 0)-member²F₄(2)′ of the infinite family ofcommutator groups²F₄(2²ⁿ⁺¹)′; thus in a strict sense not sporadic, nor of Lie type. Forn > 0 these finite simple groups coincide with the groups of Lie type²F₄(2²ⁿ⁺¹), also known asRee groups of type²F₄.

The earliest use of the termsporadic group may beBurnside (1911, p. 504) 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.)

The diagramat rightabove is based onRonan (2006, p. 247). It does not show the numerous non-sporadic simple subquotients of the sporadic groups.

Organization

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Happy Family

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Of the 26 sporadic groups, 20 can be seen inside themonster group assubgroups orquotients of subgroups (sections).These twenty have been called thehappy family byRobert Griess, and can be organized into three generations.^[10]^[b]

First generation (5 groups): the Mathieu groups

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Main article:Mathieu groups

M_n forn = 11, 12, 22, 23 and 24 are multiply transitivepermutation groups onn points. They are all subgroups of M₂₄, which is a permutation group on24 points.^[11]

Second generation (7 groups): the Leech lattice

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Third generation (8 groups): other subgroups of the Monster

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Consists of subgroups which are closely related to the Monster groupM:^[13]

B orF₂ has a double cover which is thecentralizer of an element of order 2 inM
Fi₂₄′ has a triple cover which is the centralizer of an element of order 3 inM (inconjugacy class "3A")
Fi₂₃ is a subgroup ofFi₂₄′
Fi₂₂ has a double cover which is a subgroup ofFi₂₃
The product ofTh =F₃ and a group of order 3 is the centralizer of an element of order 3 inM (in conjugacy class "3C")
The product ofHN =F₅ and a group of order 5 is the centralizer of an element of order 5 inM
The product ofHe =F₇ and a group of order 7 is the centralizer of an element of order 7 inM.
Finally, the Monster group itself is considered to be in this generation.

(This series continues further: the product ofM₁₂ and a group of order 11 is the centralizer of an element of order 11 inM.)

TheTits group, if regarded as a sporadic group, would belong in this generation: there is a subgroup S₄ ×²F₄(2)′ normalising a 2C² subgroup ofB, 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 groupFi₂₂, and thus also ofFi₂₃ andFi₂₄′, and of the Baby MonsterB.²F₄(2)′ is also a subquotient of the (pariah) Rudvalis groupRu, and has no involvements in sporadic simple groups except the ones already mentioned.

Pariahs

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Main article:Pariah group

The six exceptions areJ₁,J₃,J₄,O'N,Ru, andLy, sometimes known as thepariahs.^[14]^[15]

Table of the sporadic group orders (with Tits group)

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Group	Discoverer	^[16] Year	Generation	^[1]^[4]^[17] Order	^[18] Degree of minimal faithful Brauer character	^[19]^[20] $(a,b,ab)$ Generators	^[20]^[c] $\langle \langle a,b\mid o(z)\rangle \rangle$ Semi-presentation
M orF₁	Fischer,Griess	1973	3rd	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	$o{\bigl (}(ab)^{4}(ab^{2})^{2}{\bigr )}=50$
B orF₂	Fischer	1973	3rd	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	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=23$
Fi₂₄ orF₃₊	Fischer	1971	3rd	1,255,205,709,190,661,721,292,800 = 2²¹·3¹⁶·5²·7³·11·13·17·23·29 ≈ 1×10²⁴	8671	2A, 3E, 29	$o{\bigl (}(ab)^{3}b{\bigr )}=33$
Fi₂₃	Fischer	1971	3rd	4,089,470,473,293,004,800 = 2¹⁸·3¹³·5²·7·11·13·17·23 ≈ 4×10¹⁸	782	2B, 3D, 28	$o{\bigl (}a^{bb}(ab)^{14}{\bigr )}=5$
Fi₂₂	Fischer	1971	3rd	64,561,751,654,400 = 2¹⁷·3⁹·5²·7·11·13 ≈ 6×10¹³	78	2A, 13, 11	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=12$
Th orF₃	Thompson	1976	3rd	90,745,943,887,872,000 = 2¹⁵·3¹⁰·5³·7²·13·19·31 ≈ 9×10¹⁶	248	2, 3A, 19	$o{\bigl (}(ab)^{3}b{\bigr )}=21$
Ly	Lyons	1972	Pariah	51,765,179,004,000,000 = 2⁸·3⁷·5⁶·7·11·31·37·67 ≈ 5×10¹⁶	2480	2, 5A, 14	$o{\bigl (}ababab^{2}{\bigr )}=67$
HN orF₅	Harada,Norton	1976	3rd	273,030,912,000,000 = 2¹⁴·3⁶·5⁶·7·11·19 ≈ 3×10¹⁴	133	2A, 3B, 22	$o{\bigl (}[a,b]{\bigr )}=5$
Co₁	Conway	1969	2nd	4,157,776,806,543,360,000 = 2²¹·3⁹·5⁴·7²·11·13·23 ≈ 4×10¹⁸	276	2B, 3C, 40	$o{\bigl (}ab(abab^{2})^{2}{\bigr )}=42$
Co₂	Conway	1969	2nd	42,305,421,312,000 = 2¹⁸·3⁶·5³·7·11·23 ≈ 4×10¹³	23	2A, 5A, 28	$o{\bigl (}[a,b]{\bigr )}=4$
Co₃	Conway	1969	2nd	495,766,656,000 = 2¹⁰·3⁷·5³·7·11·23 ≈ 5×10¹¹	23	2A, 7C, 17	$o{\bigl (}(uvv)^{3}(uv)^{6}{\bigr )}=5$ ^[d]
ON orO'N	O'Nan	1976	Pariah	460,815,505,920 = 2⁹·3⁴·5·7³·11·19·31 ≈ 5×10¹¹	10944	2A, 4A, 11	$o{\bigl (}abab(b^{2}(b^{2})^{abab})^{5}{\bigr )}=5$
Suz	Suzuki	1969	2nd	448,345,497,600 = 2¹³·3⁷·5²·7·11·13 ≈ 4×10¹¹	143	2B, 3B, 13	$o{\bigl (}[a,b]{\bigr )}=15$
Ru	Rudvalis	1972	Pariah	145,926,144,000 = 2¹⁴·3³·5³·7·13·29 ≈ 1×10¹¹	378	2B, 4A, 13	$o(abab^{2})=29$
He orF₇	Held	1969	3rd	4,030,387,200 = 2¹⁰·3³·5²·7³·17 ≈ 4×10⁹	51	2A, 7C, 17	$o{\bigl (}ab^{2}abab^{2}ab^{2}{\bigr )}=10$
McL	McLaughlin	1969	2nd	898,128,000 = 2⁷·3⁶·5³·7·11 ≈ 9×10⁸	22	2A, 5A, 11	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=7$
HS	Higman,Sims	1967	2nd	44,352,000 = 2⁹·3²·5³·7·11 ≈ 4×10⁷	22	2A, 5A, 11	$o(abab^{2})=15$
J₄	Janko	1976	Pariah	86,775,571,046,077,562,880 = 2²¹·3³·5·7·11³·23·29·31·37·43 ≈ 9×10¹⁹	1333	2A, 4A, 37	$o{\bigl (}abab^{2}{\bigr )}=10$
J₃ orHJM	Janko	1968	Pariah	50,232,960 = 2⁷·3⁵·5·17·19 ≈ 5×10⁷	85	2A, 3A, 19	$o{\bigl (}[a,b]{\bigr )}=9$
J₂ orHJ	Janko	1968	2nd	604,800 = 2⁷·3³·5²·7 ≈ 6×10⁵	14	2B, 3B, 7	$o{\bigl (}[a,b]{\bigr )}=12$
J₁	Janko	1965	Pariah	175,560 = 2³·3·5·7·11·19 ≈ 2×10⁵	56	2, 3, 7	$o{\bigl (}abab^{2}{\bigr )}=19$
M₂₄	Mathieu	1861	1st	244,823,040 = 2¹⁰·3³·5·7·11·23 ≈ 2×10⁸	23	2B, 3A, 23	$o{\bigl (}ab(abab^{2})^{2}ab^{2}{\bigr )}=4$
M₂₃	Mathieu	1861	1st	10,200,960 = 2⁷·3²·5·7·11·23 ≈ 1×10⁷	22	2, 4, 23	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=8$
M₂₂	Mathieu	1861	1st	443,520 = 2⁷·3²·5·7·11 ≈ 4×10⁵	21	2A, 4A, 11	$o{\bigl (}abab^{2}{\bigr )}=11$
M₁₂	Mathieu	1861	1st	95,040 = 2⁶·3³·5·11 ≈ 1×10⁵	11	2B, 3B, 11	$o{\bigl (}[a,b]{\bigr )}=o{\bigl (}ababab^{2}{\bigr )}=6$
M₁₁	Mathieu	1861	1st	7,920 = 2⁴·3²·5·11 ≈ 8×10³	10	2, 4, 11	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=4$
T or²F₄(2)′	Tits	1964	3rd	17,971,200 = 2¹¹·3³·5²·13 ≈ 2×10⁷	104^[21]	2A, 3, 13	$o{\bigl (}[a,b]{\bigr )}=5$

Notes

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^The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups²F₄(2²ⁿ⁺¹)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.
^Conway et al. (1985, p. viii) organizes the 26 sporadic groups in likeness:
"The sporadic simple groups may be roughly sorted as the Mathieu groups, the Leech lattice groups, Fischer's 3-transposition groups, the further Monster centralizers, and the half-dozen oddments."
^Here listed aresemi-presentations from standard generators of each sporadic group. Most sporadic groups have multiple presentations & semi-presentations; the more prominent examples are listed.
^Where $u=(b^{2}(b^{2})abb)^{3}$ and $v=t(b^{2}(b^{2})t)^{2}$ with $t=abab^{3}a^{2}$ .

References

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^^a ^b ^cConway et al. (1985, p. viii)
^Griess, Jr. (1998, p. 146)
^Gorenstein, Lyons & Solomon (1998, pp. 262–302)
^^a ^bRonan (2006, pp. 244–246)
^Howlett, Rylands & Taylor (2001, p.429)
"This completes the determination of matrix generators for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group)."
^Gorenstein (1979, p.111)
^Conway et al. (1985, p.viii)
^Hartley & Hulpke (2010, p.106)
"The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if the Tits group²F₄(2)′ is counted also) which these infinite families do not include."
^Wilson et al. (1999, Sporadic groups & Exceptional groups of Lie type)
^Griess, Jr. (1982, p. 91)
^Griess, Jr. (1998, pp. 54–79)
^Griess, Jr. (1998, pp. 104–145)
^Griess, Jr. (1998, pp. 146−150)
^Griess, Jr. (1982, pp. 91−96)
^Griess, Jr. (1998, pp. 146, 150−152)
^Hiss (2003, p. 172)
Tabelle 2. Die Entdeckung der sporadischen Gruppen (Table 2. The discovery of the sporadic groups)
^Sloane (1996)
^Jansen (2005, pp. 122–123)
^Nickerson & Wilson (2011, p. 365)
^^a ^bWilson et al. (1999)
^Lubeck (2001, p. 2151)

Works cited

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Burnside, William (1911).Theory of groups of finite order(PDF) (2nd ed.). Cambridge:Cambridge University Press. pp. xxiv,1–512.doi:10.1112/PLMS/S2-7.1.1.hdl:2027/uc1.b4062919.ISBN 0-486-49575-2.MR 0069818.OCLC 54407807.S2CID 117347785.{{cite book}}:ISBN / Date incompatibility (help)

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.Bibcode:1968PNAS...61..398C.doi:10.1073/pnas.61.2.398.MR 0237634.PMC 225171.PMID 16591697.S2CID 29358882.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. Oxford:Clarendon Press. pp. xxxiii,1–252.ISBN 978-0-19-853199-9.MR 0827219.OCLC 12106933.S2CID 117473588.Zbl 0568.20001.

Gorenstein, D. (1979)."The classification of finite simple groups. I. Simple groups and local analysis".Bulletin of the American Mathematical Society. New Series.1 (1).American Mathematical Society:43–199.doi:10.1090/S0273-0979-1979-14551-8.MR 0513750.S2CID 121953006.Zbl 0414.20009.

Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1998).The classification of the finite simple groups, Number 3. Mathematical Surveys and Monographs. Vol. 40. Providence, R.I.:American Mathematical Society. pp. xiii,1–362.doi:10.1112/S0024609398255457.ISBN 978-0-8218-0391-2.MR 1490581.OCLC 6907721813.S2CID 209854856.

Griess, Jr., Robert L. (1982)."The Friendly Giant".Inventiones Mathematicae.69: 1−102.Bibcode:1982InMat..69....1G.doi:10.1007/BF01389186.hdl:2027.42/46608.MR 0671653.S2CID 123597150.Zbl 0498.20013.

Griess, Jr., Robert L. (1998).Twelve Sporadic Groups. Springer Monographs in Mathematics. Berlin:Springer-Verlag. pp. 1−169.ISBN 9783540627784.MR 1707296.OCLC 38910263.Zbl 0908.20007.

Hartley, Michael I.; Hulpke, Alexander (2010),"Polytopes Derived from Sporadic Simple Groups",Contributions to Discrete Mathematics,5 (2), Alberta, CA:University of Calgary Department of Mathematics and Statistics: 106−118,doi:10.11575/cdm.v5i2.61945,ISSN 1715-0868,MR 2791293,S2CID 40845205,Zbl 1320.51021

Hiss, Gerhard (2003)."Die Sporadischen Gruppen (The Sporadic Groups)"(PDF).Jahresber. Deutsch. Math.-Verein. (Annual Report of the German Mathematicians Association).105 (4): 169−193.ISSN 0012-0456.MR 2033760.Zbl 1042.20007. (German)