Monster group
Algebraic structure → Group theory Group theory |
---|
In the area of
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.
The
It is difficult to give a good constructive definition of the monster because of its complexity.
History
The monster was predicted by
Griess's construction showed that the monster exists. Thompson[7] 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,[8] though he never published the details. Griess, Meierfrankenfeld, and Segev 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).[9]
The monster was a culmination of the development of sporadic simple groups and can be built from any two of three 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
The smallest faithful permutation representation of the monster is on
97,239,461,142,009,186,000
= 24 · 37 · 53 · 74 · 11 · 132 · 29 · 41 · 59 · 71 ≈ 1020
points.
The monster can be realized as a
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. Alternating groups, such as A100, have permutation representations that are "small" compared to the size of the group, and all finite simple groups of Lie type, such as SL20(2), 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).
Computer construction
Martin Seysen has implemented a fast Python package named mmgroup, which claims to be the first implementation of the monster group where arbitrary operations can effectively be performed. The documentation states that multiplication of group elements takes less than 40 milliseconds on a typical modern PC, which is five orders of magnitude faster than estimated by Robert A. Wilson in 2013.[12][13][14][15] The mmgroup software package has been used to find two new maximal subgroups of the monster group.[16]
Previously, Robert A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in the field of order 2) which together generate the monster group by matrix multiplication; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices was 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.[17]
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".[18]
Wilson with collaborators found a method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work. 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
Moonshine
The monster group is one of two principal constituents in the
In this setting, the monster group is visible as the automorphism group of the
Many mathematicians, including Conway, have seen the monster as a beautiful and still mysterious object.[20] Conway said of the monster group: "There's never been any kind of explanation of why it's there, and it's obviously not there just by coincidence. It's got too many intriguing properties for it all to be just an accident."[21] Simon P. Norton, an expert on the properties of the monster group, is quoted as saying, "I can explain what Monstrous Moonshine is in one sentence, it is the voice of God."[22]
McKay's E8 observation
There are also connections between the monster and the extended Dynkin diagrams specifically between the nodes of the diagram and certain conjugacy classes in the monster, known as McKay's E8 observation.[23][24][25] This is then extended to a relation between the extended diagrams and the groups 3.Fi24′, 2.B, and M, where these are (3/2/1-fold central extensions) of the
Maximal subgroups
The monster has 46 conjugacy classes of maximal
The 46 classes of maximal subgroups of the monster are given by the following table. Previous unpublished work of Wilson et. al had purported to rule out any almost simple subgroups with non-abelian simple socles of the form U3(4), L2(8), and L2(16).[27][28][29] However, the latter was contradicted by Dietrich et al., who found a new maximal subgroup of the form U3(4). The same authors had previously found a new maximal subgroup of the form L2(13) and confirmed that there are no maximal subgroups with socle L2(8) or L2(16), thus completing the classification in the literature.[16]
Note that tables of maximal subgroups have often been found to contain subtle errors, and in particular at least two of the subgroups in the table below were incorrectly omitted from some previous lists.
No. | Structure | Order | Comments |
---|---|---|---|
1 | 2.B | 8,309,562,962,452,852,382,355,161,088,000,000 = 242.313.56.72.11.13.17.19.23.31.47 |
centralizer of an involution; contains the normalizer (47:23) × 2 of a Sylow 47-subgroup |
2 | 21+24.Co1 | 139,511,839,126,336,328,171,520,000 = 246.39.54.72.11.13.23 |
centralizer of an involution |
3 | 3.Fi24 | 7,531,234,255,143,970,327,756,800 = 222.317.52.73.11.13.17.23.29 |
normalizer of a subgroup of order 3; contains the normalizer ((29:14) × 3).2 of a Sylow 29-subgroup |
4 | 22.2E6(2):S3 | 1,836,779,512,410,596,494,540,800 = 239.310.52.72.11.13.17.19 |
normalizer of a Klein 4-group |
5 | 210+16.O+ 10(2) |
1,577,011,055,923,770,163,200 = 246.35.52.7.17.31 |
|
6 | 22+11+22.(S3 × M24) | 50,472,333,605,150,392,320 = 246.34.5.7.11.23 |
normalizer of a Klein 4-group; contains the normalizer (23:11) × S4 of a Sylow 23-subgroup |
7 | 31+12.2Suz.2 | 2,859,230,155,080,499,200 = 215.320.52.7.11.13 |
normalizer of a subgroup of order 3 |
8 | 25+10+20.(S3 × L5(2)) | 2,061,452,360,684,666,880 = 246.33.5.7.31 |
|
9 | S3 × Th | 544,475,663,327,232,000 = 216.311.53.72.13.19.31 |
normalizer of a subgroup of order 3; contains the normalizer (31:15) × S3 of a Sylow 31-subgroup |
10 | 23+6+12+18.(L3(2) × 3S6) | 199,495,389,743,677,440 = 246.34.5.7 |
|
11 | 38.O− 8(3).23 |
133,214,132,225,341,440 = 211.320.5.7.13.41 |
|
12 | (D10 × HN).2 | 5,460,618,240,000,000 = 216.36.57.7.11.19 |
normalizer of a subgroup of order 5 |
13 | (32:2 × O+ 8(3)).S4 |
2,139,341,679,820,800 = 216.315.52.7.13 |
|
14 | 32+5+10.(M11 × 2S4) | 49,093,924,366,080 = 28.320.5.11 |
|
15 | 33+2+6+6:(L3(3) × SD16) | 11,604,018,486,528 = 28.320.13 |
|
16 | 51+6:2J2:4 | 378,000,000,000 = 210.33.59.7 |
normalizer of a subgroup of order 5 |
17 | (7:3 × He):2 | 169,276,262,400 = 211.34.52.74.17 |
normalizer of a subgroup of order 7 |
18 | (A5 × A12):2 | 28,740,096,000 = 212.36.53.7.11 |
|
19 | 53+3.(2 × L3(5)) | 11,625,000,000 = 26.3.59.31 |
|
20 | (A6 × A6 × A6).(2 × S4) | 2,239,488,000 = 213.37.53 |
|
21 | (A5 × U3(8):31):2 | 1,985,679,360 = 212.36.5.7.19 |
contains the normalizer ((19:9) × A5):2 of a Sylow 19-subgroup |
22 | 52+2+4:(S3 × GL2(5)) | 1,125,000,000 = 26.32.59 |
|
23 | (L3(2) × S4(4):2).2 | 658,022,400 = 213.33.52.7.17 |
contains the normalizer ((17:8) × L3(2)).2 of a Sylow 17-subgroup |
24 | 71+4:(3 × 2S7) | 508,243,680 = 25.33.5.76 |
normalizer of a subgroup of order 7 |
25 | (52:4.22 × U3(5)).S3 | 302,400,000 = 29.33.55.7 |
|
26 | (L2(11) × M12):2 | 125,452,800 = 29.34.52.112 |
contains the normalizer (11:5 × M12):2 of a subgroup of order 11 |
27 | (A7 × (A5 × A5):22):2 | 72,576,000 = 210.34.53.7 |
|
28 | 54:(3 × 2L2(25)):22 | 58,500,000 = 25.32.56.13 |
|
29 | 72+1+2:GL2(7) | 33,882,912 = 25.32.76 |
|
30 | M11 × A6.22 | 11,404,800 = 29.34.52.11 |
|
31 | (S5 × S5 × S5):S3 | 10,368,000 = 210.34.53 |
|
32 | (L2(11) × L2(11)):4 | 1,742,400 = 26.32.52.112 |
|
33 | 132:2L2(13).4 | 1,476,384 = 25.3.7.133 |
|
34 | (72:(3 × 2A4) × L2(7)):2 | 1,185,408 = 27.33.73 |
|
35 | (13:6 × L3(3)).2 | 876,096 = 26.34.132 |
normalizer of a subgroup of order 13 |
36 | 131+2:(3 × 4S4) | 632,736 = 25.32.133 |
normalizer of a subgroup of order 13; normalizer of a Sylow 13-subgroup |
37 | U3(4):4 | 249,600 = 28.3.52.13 |
[16] |
38 | L2(71) | 178,920 = 23.32.5.7.71 |
contains the normalizer 71:35 of a Sylow 71-subgroup[30] |
39 | L2(59) | 102,660 = 22.3.5.29.59 |
contains the normalizer 59:29 of a Sylow 59-subgroup[31] |
40 | 112:(5 × 2A5) | 72,600 = 23.3.52.112 |
normalizer of a Sylow 11-subgroup. |
41 | L2(41) | 34,440 = 23.3.5.7.41 |
Norton and Wilson 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[28] |
42 | L2(29):2 | 24,360 = 23.3.5.7.29 |
[32] |
43 | 72:SL2(7) | 16,464 =24.3.73 |
this was accidentally omitted from some previous lists of 7-local subgroups |
44 | L2(19):2 | 6,840 = 23.32.5.19 |
[30] |
45 | L2(13):2 | 2,184 = 23.3.7.13 |
[16] |
46 | 41:40 | 1,640 = 23.5.41 |
normalizer of a Sylow 41-subgroup |
See also
- Supersingular prime, the prime numbers that divide the order of the monster
- Bimonster group, the wreath square of the monster group, which has a surprisingly simple presentation
Citations
- ^ Gardner 1980, pp. 20–33.
- ^ Griess 1976, pp. 113–118.
- ^ Griess 1982, pp. 1–102.
- ^ Conway 1985, pp. 513–540.
- ^ Tits 1983, pp. 105–122.
- ^ Tits 1984, pp. 491–499.
- ^ Thompson 1979, pp. 340–346.
- ^ Norton 1985, pp. 271–285.
- ^ Griess, Meierfrankenfeld & Segev 1989, pp. 567–602.
- ^ Thompson 1984, p. 443.
- ^ Wilson 2001, pp. 367–374.
- ^ Seysen, Martin. "The mmgroup API reference". Retrieved 31 July 2022.
- arXiv:2203.04223 [math.GR].
- arXiv:2002.10921 [math.GR].
- ].
- ^ a b c d e Dietrich, Lee & Popiel 2023.
- ^ Borcherds 2002, p. 1076.
- ^ Borcherds 2002, p. 1077.
- ^ Conway & Norton 1979, pp. 308–339.
- ^ Roberts 2013.
- ^ Haran 2014, 7:57.
- ^ Masters 2019.
- ^ Duncan 2008.
- ^ le Bruyn 2009.
- ^ He & McKay 2015.
- ^ Ronan 2006.
- ^ Wilson 2010, pp. 393–403.
- ^ a b Norton & Wilson 2013, pp. 943–962.
- ^ Wilson 2016, pp. 355–364.
- ^ a b Holmes & Wilson 2008, pp. 2653–2667.
- ^ Holmes & Wilson 2004, pp. 141–152.
- ^ Holmes & Wilson 2002, pp. 435–447.
Sources
- Borcherds, Richard E. (October 2002). "What is... The Monster?" (PDF). Notices of the American Mathematical Society. 49 (9).
- le Bruyn, Lieven (22 April 2009). "The monster graph and McKay's observation". neverendingbooks.
- S2CID 123340529.
- .
- Dietrich, Heiko; Lee, Melissa; Popiel, Tomasz (6 December 2023). "The maximal subgroups of the Monster". Bibcode:2023arXiv230414646D.
- Duncan, John F. (2008). "Arithmetic groups and the affine E8 Dynkin diagram". arXiv:0810.1465 [RT math. RT].
- Gardner, Martin (1980). "Mathematical games". Scientific American. Vol. 242, no. 6. pp. 20–33. JSTOR 24966339.
- 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, 1975). Boston, MA: MR 0399248.
- Griess, Robert L. (1982). "The friendly giant" (PDF). S2CID 123597150.
- Griess, Robert L.; Meierfrankenfeld, Ulrich; Segev, Yoav (1989). "A uniqueness proof for the Monster". MR 1025167.
- Haran, Brady (2014). Life, Death and the Monster (John Conway). Numberphile – via YouTube.
- arXiv:1505.06742 [AG math. AG].
- Holmes, Petra E.; Wilson, Robert A. (2002). "A new maximal subgroup of the Monster". MR 1900293.
- 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. S2CID 122913546.
- Holmes, Petra E.; Wilson, Robert A. (2008). "On subgroups of the Monster containing A5's". MR 2397402.
- Masters, Alexander (22 February 2019). "Simon Norton obituary". The Guardian.
- Norton, Simon P. (1985). "The uniqueness of the Fischer–Griess Monster". Finite groups—coming of age (Montreal, Que., 1982). Contemp. Math. Vol. 45. Providence RI: MR 0822242.
- 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). Journal of the London Mathematical Society. Second Series. 87 (3): 943–962. S2CID 7075719.
- Roberts, Siobhan (2013). Curiosities: Pursuing the Monster. Institute for Advanced Study.
- ISBN 019280722-6.
- MR 0554400.
- MR 0751155.
- Zbl 0548.20010.
- S2CID 122379975.
- MR 1859175. Archived from the originalon 2012-03-05.
- Wilson, Robert A. (2010). "New computations in the Monster". Moonshine: the first quarter century and beyond. London Math. Soc. Lecture Note Ser. Vol. 372. MR 2681789.
- Wilson, Robert A. (2016). "Is the Suzuki group Sz(8) a subgroup of the Monster?" (PDF). Bulletin of the London Mathematical Society. 48 (2): 355–364. S2CID 123219818.
Further reading
- ISBN 978-019853199-9.
- MR 1690763.
- Holmes, P. E.; S2CID 102338377.
- Holmes, Petra E. (2008). "A classification of subgroups of the Monster isomorphic to S4 and an application". MR 2408306.
- Ivanov, A.A. (2009). The Monster Group and Majorana Involutions. Cambridge tracts in mathematics. Vol. 176. Cambridge University Press. ISBN 978-052188994-0.
- 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. MR 1647423.
- Norton, Simon P.; Wilson, Robert A. (2002). "Anatomy of the Monster. II". Proceedings of the London Mathematical Society. Third Series. 84 (3): 581–598. MR 1888424.
- ISBN 978-006078940-4).
- Wilson, R. A.; Walsh, P. G.; Parker, R. A.; Linton, S. A. (1998). "Computer construction of the Monster". Journal of Group Theory. 1 (4): 307–337. .
- McKay, John; S2CID 235293875.
External links
- What is... The Monster? by Richard E. Borcherds, Notices of the American Mathematical Society, October 2002 1077
- MathWorld: Monster Group
- Atlas of Finite Group Representations: Monster group
- Scientific American June 1980 Issue: The capture of the monster: a mathematical group with a ridiculous number of elements