Monster group

In the area of abstract algebra known as group theory, the monster group M is the largest sporadic simple group; it has order
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 20 sporadic groups as subquotients. Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the happy family, and the remaining six exceptions pariahs.
It is difficult to give 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 and Robert Griess 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 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 was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s whether the monster actually existed. Griess constructed as the automorphism group of the Griess algebra, a commutative nonassociative algebra over the real numbers; he first announced his construction in Ann Arbor on 14 January 1980. In his 1982 paper, he referred to the monster as the "Friendly Giant", but this name has not been generally adopted. John Conway and Jacques Tits subsequently simplified this construction.
Griess's construction showed that the monster exists. Thompson showed that its uniqueness would follow from the existence of a faithful representation. A proof of the existence of such a representation was announced by Norton, though he never published the details. Griess, Meierfrankenfeld, and Segev gave the first complete published proof of the uniqueness of the monster.
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 faithful complex representation is which is the product of the three largest prime divisors of the order of.
The smallest faithful linear representation over any field has dimension over the field with two elements, only one less than the dimension of the smallest faithful complex representation.
The smallest faithful permutation representation of the monster is on
points.
The monster can be realized as a Galois group over the rational numbers, and as a Hurwitz group.
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 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, 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.

Computer construction

Martin Seysen implemented a fast Python package named , 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. The mmgroup software package has been used to find two new maximal subgroups of the monster group.
Previously, R.A. Wilson had found explicitly two invertible 196,882 by 196,882 matrices which together generate the monster group by matrix multiplication; this is one dimension lower than the 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.
Wilson asserts that the best description of the monster is to say,
This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".
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 be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup of the Monster is selected in which it is easy to perform calculations. The subgroup chosen is 3.2.Suz.2, where Suz is the Suzuki group. Elements of the monster are stored as words in the elements of and an extra generator. It is reasonably quick to calculate the action of one of these words on a vector in. Using this action, it is possible to perform calculations. Wilson has exhibited vectors and whose joint stabilizer is the trivial group. Thus one can calculate the order of an element of the monster by finding the smallest such that and This and similar constructions were used to find some of the non-local maximal subgroups of the monster group.

Subquotients

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.

Maximal subgroups

The monster has 46 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 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, L₂, and L₂. However, the latter was contradicted by Dietrich et al., who found a new maximal subgroup of the form U₃. The same authors had previously found a new maximal subgroup of the form L₂ and confirmed that there are no maximal subgroups with socle L₂ or L₂, thus completing the classification in the literature.

Nr.	Structure	Order	Comments
1	2^·B	8,309,562,962,452,852,382,355,161,088 ×1,000,000 = 2⁴²·3¹³·5⁶·7²·11·13·17·19·23·31·47	centralizer of an involution of class 2A; contains the normalizer of a Sylow 47 subgroup
2	2^·Co₁	139,511,839,126,336,328,171,520,000 = 2⁴⁶·3⁹·5⁴·7²·11·13·23	centralizer of an involution of class 2B
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 of a Sylow 29 subgroup
4	2^2 · 2E₆:S₃	1,836,779,512,410,596,494,540,800 = 2³⁹·3¹⁰·5²·7²·11·13·17·19	normalizer of a Klein 4 group of type 2A²
5		1,577,011,055,923,770,163,200 = 2⁴⁶·3⁵·5²·7·17·31
6	2^2+11+22.	50,472,333,605,150,392,320 = 2⁴⁶·3⁴·5·7·11·23	normalizer of a Klein 4-group; contains the normalizer of a Sylow 23 subgroup
7	3.2Suz.2	2,859,230,155,080,499,200 = 2¹⁵·3²⁰·5²·7·11·13	normalizer of a subgroup of order 3
8	2^5+10+20.	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 of a Sylow 31 subgroup
10	2^3+6+12+18.	199,495,389,743,677,440 = 2⁴⁶·3⁴·5·7
11	3^8 ·O^·2₃	133,214,132,225,341,440 = 2¹¹·3²⁰·5·7·13·41
12	.2	5,460,618,240,000,000 = 2¹⁶·3⁶·5⁷·7·11·19	normalizer of a subgroup of order 5
13	.S₄	2,139,341,679,820,800 = 2¹⁶·3¹⁵·5²·7·13
14	3²⁺⁵⁺¹⁰.	49,093,924,366,080 = 2⁸·3²⁰·5·11
15	3^3+2+6+6:	11,604,018,486,528 = 2⁸·3²⁰·13
16	5:2J₂:4	378,000,000,000 = 2¹⁰·3³·5⁹·7	normalizer of a subgroup of order 5
17	:2	169,276,262,400 = 2¹¹·3⁴·5²·7⁴·17	normalizer of a subgroup of order 7
18	:2	28,740,096,000 = 2¹²·3⁶·5³·7·11
19	5³⁺³.	11,625,000,000 = 2⁶·3·5⁹·31
20	.	2,239,488,000 = 2¹³·3⁷·5³
21	:2	1,985,679,360 = 2¹²·3⁶·5·7·19	contains the normalizer of a Sylow 19 subgroup
22	5²⁺²⁺⁴:	1,125,000,000 = 2⁶·3²·5⁹
23	× S₄.2	658,022,400 = 2¹³·3³·5²·7·17	contains the normalizer of a Sylow 17 subgroup
24	7:	508,243,680 = 2⁵·3³·5·7⁶	normalizer of a subgroup of order 7
25	.S₃	302,400,000 = 2⁹·3³·5⁵·7
26	:2	125,452,800 = 2⁹·3⁴·5²·11²	contains the normalizer of a subgroup of order 11
27	:2	72,576,000 = 2¹⁰·3⁴·5³·7
28	5⁴::2₂	58,500,000 = 2⁵·3²·5⁶·13
29	7²⁺¹⁺²:GL₂	33,882,912 = 2⁵·3²·7⁶
30	M₁₁ × A₆.2²	11,404,800 = 2⁹·3⁴·5²·11
31	:S₃	10,368,000 = 2¹⁰·3⁴·5³
32	× L₂):4	1,742,400 = 2⁶·3²·5²·11²
33	13²:2L₂.4	1,476,384 = 2⁵·3·7·13³
34	× L₂):2	1,185,408 = 2⁷·3³·7³
35	.2	876,096 = 2⁶·3⁴·13²	normalizer of a subgroup of order 13
36	13:	632,736 = 2⁵·3²·13³	normalizer of a subgroup of order 13 ; normalizer of a Sylow 13 subgroup
37	U₃:4	249,600 = 2⁸·3·5²·13
38	L₂	178,920 = 2³·3²·5·7·71	contains the normalizer 71:35 of a Sylow 71 subgroup
39	11²:	72,600 = 2³·3·5²·11²	normalizer of a Sylow 11 subgroup.
40	L₂	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 claimed that no such maximal subgroup existed
41	L₂:2	24,360 = 2³·3·5·7·29
42	7²:SL₂	16,464 =2⁴·3·7³	this was accidentally omitted from some previous lists of 7 local subgroups
43	L₂:2	6,840 = 2³·3²·5·19
44	L₂:2	2,184 = 2³·3·7·13
45	59:29	1,711 = 29·59	normalizer of a Sylow 59 subgroup; previously thought to be
46	41:40	1,640 = 2³·5·41	normalizer of a Sylow 41 subgroup

Note that tables of maximal subgroups have often been found to contain subtle errors, and in particular at least two of the subgroups in this table were incorrectly omitted from some previous lists.