In the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
known as
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest
sporadic simple 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. T ...
, having
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
2
463
205
97
611
213
3171923293141475971
= 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
≈ 8.
The
finite simple groups have been completely
classified
Classified may refer to:
General
*Classified information, material that a government body deems to be sensitive
*Classified advertising or "classifieds"
Music
*Classified (rapper) (born 1977), Canadian rapper
* The Classified, a 1980s American ro ...
. Every such group belongs to one of 18
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as
subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...
s.
Robert Griess
Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michi ...
, 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
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...
''.
History
The monster was predicted by
Bernd Fischer (unpublished, about 1973) and
Robert Griess
Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michi ...
as a simple group containing a
double cover of Fischer's
baby monster group as a
centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
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 whether the monster actually existed. Griess constructed M as the
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the
Griess algebra, a 196,884-dimensional commutative
nonassociative algebra over the real numbers; he first announced his construction in
Ann Arbor
Anne, alternatively spelled Ann, is a form of the Latin female given name Anna. This in turn is a representation of the Hebrew Hannah, which means 'favour' or 'grace'. Related names include Annie.
Anne is sometimes used as a male name in the ...
on January 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 and
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Life an ...
subsequently simplified this construction.
Griess's construction showed that the monster exists.
Thompson
Thompson may refer to:
People
* Thompson (surname)
* Thompson M. Scoon (1888–1953), New York politician
Places Australia
*Thompson Beach, South Australia, a locality
Bulgaria
* Thompson, Bulgaria, a village in Sofia Province
Canada ...
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, though he has 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).
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
In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by .
3-transposition groups
The Fischer groups are named after Bernd Fischer who discovered them ...
Fi
24, the baby monster, and the
Conway group
In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by .
The largest of the Conway groups, Co0, is the group of autom ...
Co
1.
The
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \ope ...
and the
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...
of the monster are both
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
.
Representations
The minimal degree of a
faithful complex representation is 47 × 59 × 71 = 196,883, hence is the product of the three largest
prime divisors of the order of M.
The smallest faithful linear representation over any field has dimension 196,882 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
2
4 · 3
7 · 5
3 · 7
4 · 11 · 13
2 · 29 · 41 · 59 · 71 (about 10
20)
points.
The monster can be realized as a
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, 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 A
100 and SL
20(2) are far larger, but easy to calculate with as they have "small" permutation or linear representations.
Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
s, such as A
100, have permutation representations that are "small" compared to the size of the group, and all finite simple groups of
Lie type, such as SL
20(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).
A computer construction
Robert A. Wilson has 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 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
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the
monster vertex algebra
The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by a ...
". This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".
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 3
1+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 ''g''
''i''''u'' = ''u'' and ''g''
''i''''v'' = ''v''.
This and similar constructions (in different
characteristics) have been used to find some of its non-local maximal subgroups.
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.
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
Richard Ewen Borcherds (; born 29 November 1959) is a British mathematician currently working in quantum field theory. He is known for his work in lattices, group theory, and infinite-dimensional algebras, for which he was awarded the Fields Me ...
in 1992.
In this setting, the monster group is visible as the automorphism group of the
monster module, a
vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usefu ...
, an infinite dimensional algebra containing the Griess algebra, and acts on the
monster Lie algebra
In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.
Structure
The monster Lie algebra ''m'' is a ''Z2 ...
, a
generalized Kac–Moody algebra
In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots.
Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borc ...
.
Many mathematicians, including Conway, have seen the monster as a beautiful and still mysterious object. 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."
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."
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 E
8 observation''. This is then extended to a relation between the extended diagrams
and the groups 3.Fi
24′, 2.B, and M, where these are (3/2/1-fold central extensions) of the
Fischer group
In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by .
3-transposition groups
The Fischer groups are named after Bernd Fischer who discovered them ...
,
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
In mathematics, the McKay graph of a finite-dimensional representation of a finite group is a weighted quiver encoding the structure of the representation theory of . Each node represents an irreducible representation of . If are irreducibl ...
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
In mathematics, Bring's curve (also called Bring's surface) is the curve given by the equations
:v+w+x+y+z=v^2+w^2+x^2+y^2+z^2=v^3+w^3+x^3+y^3+z^3=0.
It was named by after Erland Samuel Bring who studied a similar construction in 1786 in a Promot ...
.
Maximal subgroups
The monster has at least 44 conjugacy classes of maximal
subgroups
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
. Non-abelian simple groups of some 60
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
types are found as subgroups or as quotients of subgroups. The largest
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
represented is A
12.
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
Mark Andrew Ronan (born 1947) is Emeritus Professor of Mathematics at the University of Illinois at Chicago and Honorary Professor of Mathematics at University College London. He has lived and taught in: Germany (at the University of Braunschwe ...
, 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.
Forty-four 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 L
2(13), U
3(4), or U
3(8). 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 from some previous lists.
* 2.B centralizer of an involution; contains the normalizer (47:23) × 2 of a Sylow 47-subgroup
* 2
1+24.Co
1 centralizer of an involution
* 3.Fi
24 normalizer of a subgroup of order 3; contains the normalizer ((29:14) × 3).2 of a Sylow 29-subgroup
* 2
2.
2E
6(2
2):S
3 normalizer of a Klein 4-group
* 2
10+16.O
10+(2)
* 2
2+11+22.(M
24 × S
3) normalizer of a Klein 4-group; contains the normalizer (23:11) × S
4 of a Sylow 23-subgroup
* 3
1+12.2Suz.2 normalizer of a subgroup of order 3
* 2
5+10+20.(S
3 × L
5(2))
* S
3 × Th normalizer of a subgroup of order 3; contains the normalizer (31:15) × S
3 of a Sylow 31-subgroup
* 2
3+6+12+18.(L
3(2) × 3S
6)
* 3
8.O
8− (3).2
3
* (D
10 × HN).2 normalizer of a subgroup of order 5
* (3
2:2 × O
8+(3)).S
4
* 3
2+5+10.(M
11 × 2S
4)
* 3
3+2+6+6:(L
3(3) × SD
16)
* 5
1+6:2J
2:4 normalizer of a subgroup of order 5
* (7:3 × He):2 normalizer of a subgroup of order 7
* (A
5 × A
12):2
* 5
3+3.(2 × L
3(5))
* (A
6 × A
6 × A
6).(2 × S
4)
* (A
5 × U
3(8):3
1):2 contains the normalizer ((19:9) × A
5):2 of a Sylow 19-subgroup
* 5
2+2+4:(S
3 × GL
2(5))
* (L
3(2) × S
4(4):2).2 contains the normalizer ((17:8) × L
3(2)).2 of a Sylow 17-subgroup
* 7
1+4:(3 × 2S
7) normalizer of a subgroup of order 7
* (5
2:4.2
2 × U
3(5)).S
3
* (L
2(11) × M
12):2 contains the normalizer (11:5 × M
12):2 of a subgroup of order 11
* (A
7 × (A
5 × A
5):2
2):2
* 5
4:(3 × 2L
2(25)):2
2
* 7
2+1+2:GL
2(7)
* M
11 × A
6.2
2
* (S
5 × S
5 × S
5):S
3
* (L
2(11) × L
2(11)):4
* 13
2:2L
2(13).4
* (7
2:(3 × 2A
4) × L
2(7)):2
* (13:6 × L
3(3)).2 normalizer of a subgroup of order 13
* 13
1+2:(3 × 4S
4) normalizer of a subgroup of order 13; normalizer of a Sylow 13-subgroup
* L
2(71) contains the normalizer 71:35 of a Sylow 71-subgroup
* L
2(59) contains the normalizer 59:29 of a Sylow 59-subgroup
* 11
2:(5 × 2A
5) normalizer of a Sylow 11-subgroup.
* L
2(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
* L
2(29):2
* 7
2:SL
2(7) this was accidentally omitted from some previous lists of 7-local subgroups
* L
2(19):2
* 41:40 normalizer of a Sylow 41-subgroup
See also
*
Supersingular prime, the prime numbers that divide the order of the monster
Citations
Sources
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Further reading
*
*
*
*
*
*
*
* published in the US by HarperCollins as ''Symmetry'', ).
*
*
External links
''What is... The Monster?''by
Richard E. Borcherds, Notices of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
, October 2002 1077
MathWorld: Monster GroupAtlas of Finite Group Representations: Monster groupScientific American June 1980 Issue: The capture of the monster: a mathematical group with a ridiculous number of elements
{{DEFAULTSORT:Monster Group
Moonshine theory
Sporadic groups