Monster group

In the area of modern algebra known as group theory, the Monster group M (also known as the Fischer–Griess Monster, or the Friendly Giant) is the largest sporadic simple group, having order

2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71

= 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

≈ 8×10⁵³.

The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The Monster group contains all but six of the other sporadic groups as subquotients. Robert Griess has called these 6 exceptions pariahs, and refers to the other 20 as the happy family.

It is difficult to make a good constructive definition of the Monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in Scientific American.

History

The Monster was predicted by Bernd Fischer (unpublished, about 1973) and Robert Griess (1976) as a simple group containing a double cover of Fischer's Baby Monster group as a centralizer of an involution. Within a few months the order of M was found by Griess using the Thompson order formula, and Fischer, Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the Thompson group and the Harada–Norton group. The character table of the Monster, a 194-by-194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s that the Monster actually existed. Griess (1982) constructed M as the automorphism group of the Griess algebra, a 196,884-dimensional commutative nonassociative algebra; he first announced his construction in Ann Arbor Jan. 14,1980. In his 1982 paper he referred to the Monster as the Friendly Giant, but this name has not been generally adopted. John Conway (1985) and Jacques Tits (1984, 1985) subsequently simplified this construction.

Griess's construction showed that the Monster existed. Thompson (1979) showed that its uniqueness (as a simple group satisfying certain conditions coming from the classification of finite simple groups) would follow from the existence of a 196,883-dimensional faithful representation. A proof of the existence of such a representation was announced by Norton (1985), though he has never published the details. Griess, Meierfrankenfeld & Segev (1989) gave the first complete published proof of the uniqueness of the Monster (more precisely, they showed that a group with the same centralizers of involutions as the Monster is isomorphic to the Monster).

The Monster was a culmination of a development of sporadic simple groups and can be built from any 2 of 3 subquotients: the Fischer group Fi₂₄, the Baby Monster, and the Conway group Co₁.

The Schur multiplier and the outer automorphism group of the Monster are both trivial.

Representations

The minimal degree of a faithful complex representation is 196,883, which is the product of the 3 largest prime divisors of the order of M. The smallest faithful linear representation over any field has dimension 196,882 over the field with 2 elements, only 1 less than the dimension of the smallest faithful complex representation.

The smallest faithful permutation representation of the Monster is on 2⁴ · 3⁷ · 5³ · 7⁴ · 11 · 13² · 29 · 41 · 59 · 71 (about 10²⁰) points.

The Monster can be realized as a Galois group over the rational numbers (Thompson 1984, p. 443), and as a Hurwitz group (Wilson 2004).

The Monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of "small" representations. For example, the simple groups A₁₀₀ and SL₂₀(2) are far larger, but easy to calculate with as they have "small" permutation or linear representations. The alternating groups have permutation representations that are "small" compared to the size of the group, and all finite simple groups of Lie type have linear representations that are "small" compared to the size of the group. All sporadic groups other than the Monster also have linear representations small enough that they are easy to work with on a computer (the next hardest case after the monster is the baby Monster, with a representation of dimension 4370).

A computer construction

Robert A. Wilson has found explicitly (with the aid of a computer) two 196,882 by 196,882 matrices (with elements in the field of order 2) which together generate the Monster group; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices is possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.

Wilson asserts that the best description of the Monster is to say, "It is the automorphism group of the monster vertex algebra". This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".^[1]

Wilson with collaborators has found a method of performing calculations with the Monster that is considerably faster. Let V be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup H (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is 3¹⁺¹².2.Suz.2, where Suz is the Suzuki group. Elements of the Monster are stored as words in the elements of H and an extra generator T. It is reasonably quick to calculate the action of one of these words on a vector in V. Using this action, it is possible to perform calculations (such as the order of an element of the Monster). Wilson has exhibited vectors u and v whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element g of the monster by finding the smallest i > 0 such that gⁱu = u and gⁱv = v.

This and similar constructions (in different characteristics) have been used to find some of its non-local maximal subgroups.

Moonshine

The Monster group is one of two principal constituents in the Monstrous moonshine conjecture by Conway and Norton, which relates discrete and non-discrete mathematics and was finally proved by Richard Borcherds in 1992.

In this setting, the Monster group is visible as the automorphism group of the Monster module, a vertex operator algebra, an infinite dimensional algebra containing the Griess algebra, and acts on the Monster Lie algebra, a generalized Kac–Moody algebra.

McKay's E₈ observation

There are also connections between the Monster and the extended Dynkin diagrams ${\tilde {E}}_{8},$ specifically between the nodes of the diagram and certain conjugacy classes in the Monster, known as McKay's E₈ observation.^[2]^[3]^[4] This is then extended to a relation between the extended diagrams ${\tilde {E}}_{6},{\tilde {E}}_{7},{\tilde {E}}_{8}$ and the groups 3.Fi₂₄', 2.B, and M, where these are (3/2/1-fold central extensions) of the Fischer group, Baby Monster group, and Monster. These are the sporadic groups associated with centralizers of elements of type 1A, 2A, and 3A in the Monster, and the order of the extension corresponds to the symmetries of the diagram. See ADE classification: trinities for further connections (of McKay correspondence type), including (for the Monster) with the rather small simple group PSL(2,11) and with the 120 tritangent planes of a canonic sextic curve of genus 4 known as Bring's curve.

Maximal subgroups

Diagram of the 26 sporadic simple groups, showing subquotient relationships. The sporadic groups not involved in any larger one are circled.

The Monster has at least 44 conjugacy classes of maximal subgroups. Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largest alternating group represented is A₁₂. The Monster contains 20 of the 26 sporadic groups as subquotients. This diagram, based on one in the book Symmetry and the Monster by Mark Ronan, shows how they fit together. The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.

44 of the classes of maximal subgroups of the Monster are given by the following list, which is (as of 2016) believed to be complete except possibly for almost simple subgroups with non-abelian simple socles of the form L₂(13), U₃(4), or U₃(8) (Wilson 2010), (Norton & Wilson 2013), (Wilson 2016). However, tables of maximal subgroups have often been found to contain subtle errors, and in particular at least two of the subgroups on the list below were incorrectly omitted in some previous lists.

2.B Centralizer of an involution; contains the normalizer (47:23) × 2 of a Sylow 47-subgroup.
2¹⁺²⁴.Co₁ Centralizer of an involution.
3.Fi₂₄ Normalizer of a subgroup of order 3; contains the normalizer ((29:14) × 3).2 of a Sylow 29-subgroup.
2².²E₆(2²):S₃ Normalizer of a Klein 4-group.
2¹⁰⁺¹⁶.O₁₀⁺(2)
2^2+11+22.(M₂₄ × S₃) Normalizer of a Klein 4-group; contains the normalizer (23:11) × S₄ of a Sylow 23-subgroup.
3¹⁺¹².2Suz.2 Normalizer of a subgroup of order 3.
2^5+10+20.(S₃ × L₅(2))
S₃ × Th Normalizer of a subgroup of order 3; contains the normalizer (31:15) × S₃ of a Sylow 31-subgroup.
2^3+6+12+18.(L₃(2) × 3S₆)
3⁸.O₈⁻(3).2₃
(D₁₀ × HN).2 Normalizer of a subgroup of order 5.
(3²:2 × O₈⁺(3)).S₄
3²⁺⁵⁺¹⁰.(M₁₁ × 2S₄)
3^3+2+6+6:(L₃(3) × SD₁₆)
5¹⁺⁶:2J₂:4 Normalizer of a subgroup of order 5.
(7:3 × He):2 Normalizer of a subgroup of order 7.
(A₅ × A₁₂):2
5³⁺³.(2 × L₃(5))
(A₆ × A₆ × A₆).(2 × S₄)
(A₅ × U₃(8):3₁):2 Contains the normalizer ((19:9) × A₅):2 of a Sylow 19-subgroup.
5²⁺²⁺⁴:(S₃ × GL₂(5))
(L₃(2) × S₄(4):2).2 Contains the normalizer ((17:8) × L₃(2)).2 of a Sylow 17-subgroup.
7¹⁺⁴:(3 × 2S₇) Normalizer of a subgroup of order 7.
(5²:4.2² × U₃(5)).S₃
(L₂(11) × M₁₂):2 Contains the normalizer (11:5 × M₁₂):2 of a subgroup of order 11.
(A₇ × (A₅ × A₅):2²):2
5⁴:(3 × 2L₂(25)):2₂
7²⁺¹⁺²:GL₂(7)
M₁₁ × A₆.2²
(S₅ × S₅ × S₅):S₃
(L₂(11) × L₂(11)):4
13²:2L₂(13).4
(7²:(3 × 2A₄) × L₂(7)):2
(13:6 × L₃(3)).2 Normalizer of a subgroup of order 13.
13¹⁺²:(3 × 4S₄) Normalizer of a subgroup of order 13; normalizer of a Sylow 13-subgroup.
L₂(71) Holmes & Wilson (2008) Contains the normalizer 71:35 of a Sylow 71-subgroup.
L₂(59) Holmes & Wilson (2004) Contains the normalizer 59:29 of a Sylow 59-subgroup.
11²:(5 × 2A₅) Normalizer of a Sylow 11-subgroup.
L₂(41) Norton & Wilson (2013) found a maximal subgroup of this form; due to a subtle error pointed out by Zavarnitsine some previous lists and papers stated that no such maximal subgroup existed.
L₂(29):2 Holmes & Wilson (2002)
7²:SL₂(7) This was accidentally omitted on some previous lists of 7-local subgroups.
L₂(19):2 Holmes & Wilson (2008)
41:40 Normalizer of a Sylow 41-subgroup.

Notes

^ What is… The Monster? by Richard E. Borcherds, Notices of the American Mathematical Society, October 2002 1077
^ Arithmetic groups and the affine E₈ Dynkin diagram, by John F. Duncan, in Groups and symmetries: from Neolithic Scots to John McKay
^ le Bruyn, Lieven (22 April 2009), the monster graph and McKay's observation
^ He, Yang-Hui; McKay, John (2015-05-25). "Sporadic and Exceptional". arXiv:1505.06742 [math.AG].

References

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339.
Conway, John Horton (1985), "A simple construction for the Fischer–Griess monster group", Inventiones Mathematicae, 79 (3): 513–540, doi:10.1007/BF01388521, ISSN 0020-9910, MR 0782233
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England 1985.
Griess, Robert L. (1976), "The structure of the monster simple group", in Scott, W. Richard; Gross, Fletcher (eds.), Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Boston, MA: Academic Press, pp. 113–118, ISBN 978-0-12-633650-4, MR 0399248
Griess, Robert L. (1982), "The friendly giant", Inventiones Mathematicae, 69 (1): 1–102, doi:10.1007/BF01389186, ISSN 0020-9910, MR 0671653
Griess, Robert L; Meierfrankenfeld, Ulrich; Segev, Yoav (1989), "A uniqueness proof for the Monster", Annals of Mathematics. Second Series, 130 (3): 567–602, doi:10.2307/1971455, ISSN 0003-486X, JSTOR 1971455, MR 1025167
Harada, Koichiro (2001), "Mathematics of the Monster", Sugaku Expositions, 14 (1): 55–71, ISSN 0898-9583, MR 1690763
Holmes, P. E.; Wilson, R. A. (2002), "A new maximal subgroup of the Monster", Journal of Algebra, 251 (1): 435–447, doi:10.1006/jabr.2001.9037, ISSN 0021-8693, MR 1900293
P. E. Holmes and R. A. Wilson, A computer construction of the Monster using 2-local subgroups, J. London Math. Soc. 67 (2003), 346–364.
Holmes, Petra E.; Wilson, Robert A. (2004), "PSL₂(59) is a subgroup of the Monster", Journal of the London Mathematical Society. Second Series, 69 (1): 141–152, doi:10.1112/S0024610703004915, ISSN 0024-6107, MR 2025332
Holmes, Petra E.; Wilson, Robert A. (2008), "On subgroups of the Monster containing A₅'s", Journal of Algebra, 319 (7): 2653–2667, doi:10.1016/j.jalgebra.2003.11.014, ISSN 0021-8693, MR 2397402
Holmes, P. E. (2008), "A classification of subgroups of the Monster isomorphic to S₄ and an application", Journal of Algebra, 319 (8): 3089–3099, doi:10.1016/j.jalgebra.2004.01.031, ISSN 0021-8693, MR 2408306
Ivanov, A. A., The Monster Group and Majorana Involutions, Cambridge tracts in mathematics, vol. 176, Cambridge University Press, ISBN 978-0-521-88994-0
S. A. Linton, R. A. Parker, P. G. Walsh and R. A. Wilson, Computer construction of the Monster, J. Group Theory 1 (1998), 307–337.
Norton, Simon P. (1985), "The uniqueness of the Fischer–Griess Monster", Finite groups—coming of age (Montreal, Que., 1982), Contemp. Math., vol. 45, Providence, R.I.: American Mathematical Society, pp. 271–285, doi:10.1090/conm/045/822242, MR 0822242
Norton, Simon P.; Wilson, Robert A. (2002), "Anatomy of the Monster. II", Proceedings of the London Mathematical Society. Third Series, 84 (3): 581–598, doi:10.1112/S0024611502013357, ISSN 0024-6115, MR 1888424
Norton, Simon P. (1998), "Anatomy of the Monster. I", The atlas of finite groups: ten years on (Birmingham, 1995), London Math. Soc. Lecture Note Ser., vol. 249, Cambridge University Press, pp. 198–214, doi:10.1017/CBO9780511565830.020, ISBN 978-0-521-57587-4, MR 1647423
Norton, Simon P.; Wilson, Robert A. (2013), "A correction to the 41-structure of the Monster, a construction of a new maximal subgroup L2(41) and a new Moonshine phenomenon" (PDF), J. Lond. Math. Soc. (2), 87 (3): 943–962
M. Ronan, Symmetry and the Monster, Oxford University Press, 2006, ISBN 0-19-280722-6 (concise introduction for the lay reader).
M. du Sautoy, Finding Moonshine, Fourth Estate, 2008, ISBN 978-0-00-721461-7 (another introduction for the lay reader; published in the US by HarperCollins as Symmetry, ISBN 978-0-06-078940-4).
Thompson, John G. (1979), "Uniqueness of the Fischer-Griess monster", The Bulletin of the London Mathematical Society, 11 (3): 340–346, doi:10.1112/blms/11.3.340, ISSN 0024-6093, MR 0554400
Thompson, John G. (1984), "Some finite groups which appear as Gal L/K, where K ⊆ Q(μ_n)", Journal of Algebra, 89 (2): 437–499, doi:10.1016/0021-8693(84)90228-X, MR 0751155
Tits, Jacques (1984), "On R. Griess' "friendly giant"", Inventiones Mathematicae, 78 (3): 491–499, doi:10.1007/BF01388446, ISSN 0020-9910, MR 0768989
Tits, Jacques (1985), "Le Monstre (d'après R. Griess, B. Fischer et al.)", Astérisque (121): 105–122, ISSN 0303-1179, MR 0768956
Wilson, Robert A. (2010), "New computations in the Monster", Moonshine: the first quarter century and beyond, London Math. Soc. Lecture Note Ser., vol. 372, Cambridge University Press, pp. 393–403, ISBN 978-0-521-10664-1, MR 2681789
Wilson, R. A. (2001), "The Monster is a Hurwitz group", Journal of Group Theory, 4 (4): 367–374, doi:10.1515/jgth.2001.027, MR1859175
Wilson, Robert A. (2016), "Is the Suzuki group Sz(8) a subgroup of the Monster?", Bull. Lond. Math. Soc., 48 (2): 355–364, MR 3483073

External links

[1] What is… The Monster? by Richard E. Borcherds, Notices of the American Mathematical Society, October 2002 1077

[2] Arithmetic groups and the affine E₈ Dynkin diagram, by John F. Duncan, in Groups and symmetries: from Neolithic Scots to John McKay

[monster-3] Bruyn, Lieven (22 April 2009), the monster graph and McKay's observation

[4] He, Yang-Hui; McKay, John (2015-05-25). "Sporadic and Exceptional". arXiv:1505.06742 [math.AG].

[1]

[2]

[3]

[4]