Sporadic group

In 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[lower-alpha 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,[1] 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.[2]

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:[1][3][4]

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 (M11), M₁₂ (M12), M₂₂ (M22), M₂₃ (M23), M₂₄ (M24)
Janko groups J₁ (J1), J₂ or HJ (J2), J₃ or HJM (J3), J₄ (J4)
Conway groups Co₁ (Co1), Co₂ (Co2), Co₃ (Co3)
Fischer groups Fi₂₂ (Fi22), Fi₂₃ (Fi23), Fi₂₄′ or F₃₊ (Fi24)
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 et al. (1985), 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 et al. (1999), 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).

An exception found in the classification of sporadic groups within finite simple groups is the Tits group T, that is sometimes also considered as being 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.[1] In some other sources, the Tits group is regarded as neither sporadic nor of Lie type.[lower-alpha 2] The Tits group is the (n = 0)-member ²F₄(2)′ of the infinite family of commutator groups ²F₄(2²ⁿ⁺¹)′; thus by definition not sporadic. 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.

The earliest use of the term sporadic group may be Burnside (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."

The diagram at right is based on Ronan (2006, p. 247). It does not show the numerous non-sporadic simple subquotients of the sporadic groups.

Organization

Happy Family

Of the 26 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.[5][lower-alpha 3]

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.[6]

Second generation (7 groups): the Leech lattice

All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:[7]

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:[8]

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.[9][10]

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

Group	Discoverer	[11] Year	Generation	[4][12] Order		[1][4] Factorized order	[13] Minimal faithful Brauer character degree	[14][15] $(a,b,ab)$ Generators	[15][lower-alpha 4] $\langle \langle a,b\mid o(z)\rangle \rangle$ Semi-presentation
M or F₁	Fischer, Griess	1973	3rd	808017424794512875886459904961710757005754368000000000	≈ 8×10⁵³	2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71	196883	2A, 3B, 29	$o{\bigl (}(ab)^{4}(ab^{2})^{2}{\bigr )}=50$
B or F₂	Fischer	1973	3rd	4154781481226426191177580544000000	≈ 4×10³³	2⁴¹ · 3¹³ · 5⁶ · 7² · 11 · 13 · 17 · 19 · 23 · 31 · 47	4371	2C, 3A, 55	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=23$
Fi₂₄ or F₃₊	Fischer	1971	3rd	1255205709190661721292800	≈ 1×10²⁴	2²¹ · 3¹⁶ · 5² · 7³ · 11 · 13 · 17 · 23 · 29	8671	2A, 3E, 29	$o{\bigl (}(ab)^{3}b{\bigr )}=33$
Fi₂₃	Fischer	1971	3rd	4089470473293004800	≈ 4×10¹⁸	2¹⁸ · 3¹³ · 5² · 7 · 11 · 13 · 17 · 23	782	2B, 3D, 28	$o{\bigl (}a^{bb}(ab)^{14}{\bigr )}=5$
Fi₂₂	Fischer	1971	3rd	64561751654400	≈ 6×10¹³	2¹⁷ · 3⁹ · 5² · 7 · 11 · 13	78	2A, 13, 11	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=12$
Th or F₃	Thompson	1976	3rd	90745943887872000	≈ 9×10¹⁶	2¹⁵ · 3¹⁰ · 5³ · 7² · 13 · 19 · 31	248	2, 3A, 19	$o{\bigl (}(ab)^{3}b{\bigr )}=21$
Ly	Lyons	1972	Pariah	51765179004000000	≈ 5×10¹⁶	2⁸ · 3⁷ · 5⁶ · 7 · 11 · 31 · 37 · 67	2480	2, 5A, 14	$o{\bigl (}ababab^{2}{\bigr )}=67$
HN or F₅	Harada, Norton	1976	3rd	273030912000000	≈ 3×10¹⁴	2¹⁴ · 3⁶ · 5⁶ · 7 · 11 · 19	133	2A, 3B, 22	$o{\bigl (}[a,b]{\bigr )}=5$
Co₁	Conway	1969	2nd	4157776806543360000	≈ 4×10¹⁸	2²¹ · 3⁹ · 5⁴ · 7² · 11 · 13 · 23	276	2B, 3C, 40	$o{\bigl (}ab(abab^{2})^{2}{\bigr )}=42$
Co₂	Conway	1969	2nd	42305421312000	≈ 4×10¹³	2¹⁸ · 3⁶ · 5³ · 7 · 11 · 23	23	2A, 5A, 28	$o{\bigl (}[a,b]{\bigr )}=4$
Co₃	Conway	1969	2nd	495766656000	≈ 5×10¹¹	2¹⁰ · 3⁷ · 5³ · 7 · 11 · 23	23	2A, 7C, 17	$o{\bigl (}(uvv)^{3}(uv)^{6}{\bigr )}=5$ [lower-alpha 5]
ON or O'N	O'Nan	1976	Pariah	460815505920	≈ 5×10¹¹	2⁹ · 3⁴ · 5 · 7³ · 11 · 19 · 31	10944	2A, 4A, 11	$o{\bigl (}abab(b^{2}(b^{2})^{abab})^{5}{\bigr )}=5$
Suz	Suzuki	1969	2nd	448345497600	≈ 4×10¹¹	2¹³ · 3⁷ · 5² · 7 · 11 · 13	143	2B, 3B, 13	$o{\bigl (}[a,b]{\bigr )}=15$
Ru	Rudvalis	1972	Pariah	145926144000	≈ 1×10¹¹	2¹⁴ · 3³ · 5³ · 7 · 13 · 29	378	2B, 4A, 13	$o(abab^{2})=29$
He or F₇	Held	1969	3rd	4030387200	≈ 4×10⁹	2¹⁰ · 3³ · 5² · 7³ · 17	51	2A, 7C, 17	$o{\bigl (}ab^{2}abab^{2}ab^{2}{\bigr )}=10$
McL	McLaughlin	1969	2nd	898128000	≈ 9×10⁸	2⁷ · 3⁶ · 5³ · 7 · 11	22	2A, 5A, 11	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=7$
HS	Higman, Sims	1967	2nd	44352000	≈ 4×10⁷	2⁹ · 3² · 5³ · 7 · 11	22	2A, 5A, 11	$o(abab^{2})=15$
J₄	Janko	1976	Pariah	86775571046077562880	≈ 9×10¹⁹	2²¹ · 3³ · 5 · 7 · 11³ · 23 · 29 · 31 · 37 · 43	1333	2A, 4A, 37	$o{\bigl (}abab^{2}{\bigr )}=10$
J₃ or HJM	Janko	1968	Pariah	50232960	≈ 5×10⁷	2⁷ · 3⁵ · 5 · 17 · 19	85	2A, 3A, 19	$o{\bigl (}[a,b]{\bigr )}=9$
J₂ or HJ	Janko	1968	2nd	604800	≈ 6×10⁵	2⁷ · 3³ · 5² · 7	14	2B, 3B, 7	$o{\bigl (}[a,b]{\bigr )}=12$
J₁	Janko	1965	Pariah	175560	≈ 2×10⁵	2³ · 3 · 5 · 7 · 11 · 19	56	2, 3, 7	$o{\bigl (}abab^{2}{\bigr )}=19$
T (or $2 F 4 (2)'$ )	Tits	1964	3rd	17971200	≈ 2×10⁷	2¹¹ · 3³ · 5² · 13	104[16]	2A, 3, 13	$o{\bigl (}[a,b]{\bigr )}=5$
M₂₄	Mathieu	1861	1st	244823040	≈ 2×10⁸	2¹⁰ · 3³ · 5 · 7 · 11 · 23	23	2B, 3A, 23	$o{\bigl (}ab(abab^{2})^{2}ab^{2}{\bigr )}=4$
M₂₃	Mathieu	1861	1st	10200960	≈ 1×10⁷	2⁷ · 3² · 5 · 7 · 11 · 23	22	2, 4, 23	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=8$
M₂₂	Mathieu	1861	1st	443520	≈ 4×10⁵	2⁷ · 3² · 5 · 7 · 11	21	2A, 4A, 11	$o{\bigl (}abab^{2}{\bigr )}=11$
M₁₂	Mathieu	1861	1st	95040	≈ 1×10⁵	2⁶ · 3³ · 5 · 11	11	2B, 3B, 11	$o{\bigl (}[a,b]{\bigr )}=o{\bigl (}ababab^{2}{\bigr )}=6$
M₁₁	Mathieu	1861	1st	7920	≈ 8×10³	2⁴ · 3² · 5 · 11	10	2, 4, 11	$o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=4$

Notes

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.
For example, in Eric W. Weisstein, "Tits Group", MathWorld there is a link from the Tits group to "Sporadic Group", as opposed to in Eric W. Weisstein, "Sporadic Group", MathWorld, where the Tits group is not listed among the 26 sporadic groups. Both sources checked on 2018-05-26.
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 are semi-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

Conway et al. (1985, p. viii)
Griess, Jr. (1998, p. 146)
Gorenstein, Lyons & Solomon (1998, pp. 262–302)
Ronan (2006, pp. 244–246)
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 (2015)
Jansen (2005, pp. 122–123)
Nickerson & Wilson (2011, p. 365)
Wilson et al. (1999)
Lubeck (2001, p. 2151)

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. hdl:2027/uc1.b4062919. ISBN 0-486-49575-2. MR 0069818. OCLC 54407807. S2CID 117347785.

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.; 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.

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)

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. MR 2153793. S2CID 121362819. Zbl 1089.20006.

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. MR 1837968. S2CID 122060727. Zbl 1004.20003.

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. MR 2172713. S2CID 13100616. Zbl 1087.20025.

Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. vii, 1–255. doi:10.1007/s00283-008-9007-9. ISBN 978-0-19-280722-9. MR 2215662. OCLC 180766312. Zbl 1113.00002.

Sloane, N.J.A., ed. (2015). "Orders of sporadic simple groups (A001228)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Wilson, R.A (1998). "Chapter: An Atlas of Sporadic Group Representations" (PDF). 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. S2CID 59394831. Zbl 0914.20016.

Wilson, R.A.; Parker, R.A.; Nickerson, S.J.; Bray, J.N. (1999). "ATLAS: Sporadic Groups". ATLAS of Finite Group Representations.

External links

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[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.

[6] For example, in Eric W. Weisstein, "Tits Group", MathWorld there is a link from the Tits group to "Sporadic Group", as opposed to in Eric W. Weisstein, "Sporadic Group", MathWorld, where the Tits group is not listed among the 26 sporadic groups. Both sources checked on 2018-05-26.

[8] 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."

[19] Here listed are semi-presentations from standard generators of each sporadic group. Most sporadic groups have multiple presentations & semi-presentations; the more prominent examples are listed.

[20] Where $u=(b^{2}(b^{2})abb)^{3}$ and $v=t(b^{2}(b^{2})t)^{2}$ with $t=abab^{3}a^{2}$ .

[Conway-2] Conway et al. (1985, p. viii)

[3] Griess, Jr. (1998, p. 146)

[4] Gorenstein, Lyons & Solomon (1998, pp. 262–302)

[Ronan-5] Ronan (2006, pp. 244–246)

[7] Griess, Jr. (1982, p. 91)

[9] Griess, Jr. (1998, pp. 54–79)

[10] Griess, Jr. (1998, pp. 104–145)

[11] Griess, Jr. (1998, pp. 146−150)

[12] Griess, Jr. (1982, pp. 91−96)

[13] Griess, Jr. (1998, pp. 146, 150−152)

[14] Hiss (2003, p. 172)
Tabelle 2. Die Entdeckung der sporadischen Gruppen (Table 2. The discovery of the sporadic groups)

[15] Sloane (2015)

[16] Jansen (2005, pp. 122–123)

[17] Nickerson & Wilson (2011, p. 365)

[Wilsonetal-18] Wilson et al. (1999)

[21] Lubeck (2001, p. 2151)