In the area ofabstract algebra known asgroup theory, themonster group M (also known as theFischer–Griess monster, thefriendly giant, or simply theMonster) is the largestsporadic simple group; it hasorder

  808017424794512875886459904961710757005754368000000000

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

= 32! · 10! · ( 4! )² · 2 · 7 · 13 · 41 · 47 · 59 · 71

≈ 8.0802 × 10⁵³ .

Thefinitesimple groups have been completelyclassified. 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 (including itself) assubquotients.Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups thehappy family, and the remaining six exceptionspariahs.

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 1980Mathematical Games column inScientific American.^[1]

The monster was predicted byBernd Fischer (unpublished, about 1973) andRobert Griess^[2] as a simple group containing adouble cover of Fischer'sbaby monster group as acentralizer of aninvolution. Within a few months, the order ofM was found by Griess using theThompson order formula, and Fischer,Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: theThompson group and theHarada–Norton group. Thecharacter table of the monster, a194 × 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^[3] constructedM as theautomorphism group of theGriess algebra, a196883 dimensional commutativenonassociative algebra over the real numbers; he first announced his construction inAnn 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^[4] andJacques Tits^[5][6] subsequently simplified this construction.

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 a196883 dimensionalfaithful representation. A proof of the existence of such a representation was announced byNorton,^[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: TheFischer group Fi₂₄, thebaby monster, and theConway group Co₁.

TheSchur multiplier and theouter automorphism group of the monster are bothtrivial.

The minimal degree of afaithful complex representation is 47 × 59 × 71 =196883 , which is the product of the three largestprime divisors of the order ofM.The smallest faithful linear representation over any field has dimension196882 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

   97239461142009186000

= 2⁴ · 3⁷ · 5³ · 7⁴ · 11 · 13² · 29 · 41 · 59 · 71 ≈ 10²⁰

points.

The monster can be realized as aGalois group over therational numbers,^[10] and as aHurwitz group.^[11]

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 ofLie 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 thebaby monster, with a representation of dimension4370).

Martin Seysen (2022) implemented a fastPython package namedmmgroup, 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 byRobert 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,R.A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements inthe field of order 2) which togethergenerate the monster group by matrix multiplication; this is one dimension lower than the196883 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 theautomorphism group of themonster vertex algebra".

This is not much help however, because nobody has found a "really simple and natural construction of themonster 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. LetV be a 196,882 dimensional vector space over the field with 2 elements. A large subgroupH (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroupH chosen is 3¹⁺¹².2.Suz.2, where Suz is theSuzuki group. Elements of the monster are stored as words in the elements ofH and an extra generatorT. It is reasonably quick to calculate the action of one of these words on a vector inV. Using this action, it is possible to perform calculations (such as the order of an element of the monster). Wilson has exhibited vectorsu andv whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an elementg of the monster by finding the smallest k > 0  such that g^k u = u  and g^k v = v . This and similar constructions (in differentcharacteristics) were used to find some of the non-local maximal subgroups of the monster group.

The monster contains 20 of the 26 sporadic groups as subquotients. This diagram, based on one in the bookSymmetry and the Monster byMark Ronan, shows how they fit together.^[19] 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.

The monster has 46 conjugacy classes of maximalsubgroups.^[16] Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largestalternating 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 simplesocles of the form U₃(4), L₂(8), and L₂(16).^[20][21][22] 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 this table were incorrectly omitted from some previous lists.

There are also connections between the monster and the extendedDynkin diagrams${\displaystyle {\tilde{E}}_8}$ specifically between the nodes of the diagram and certain conjugacy classes in the monster, known asMcKay's E₈ observation.^[25][26][27] This is then extended to a relation between the extended diagrams${\displaystyle {\tilde{E}}_6,{\tilde{E}}_7,{\tilde{E}}_8}$ and the groups 3.Fi₂₄′, 2.B, and M, where these are (3/2/1-fold central extensions) of theFischer group,baby monster group, and monster. These are thesporadic 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. SeeADE classification: trinities for further connections (ofMcKay correspondence type), including (for the monster) with the rather small simple groupPSL(2,11) and with the 120 tritangent planes of a canonic sextic curve of genus 4 known asBring's curve.

The monster group is one of two principal constituents in themonstrous moonshine conjecture by Conway and Norton,^[28] which relates discrete and non-discrete mathematics and was finally proved byRichard Borcherds in 1992.

In this setting, the monster group is visible as the automorphism group of themonster module, avertex operator algebra, an infinite dimensional algebra containing the Griess algebra, and acts on themonster Lie algebra, ageneralized Kac–Moody algebra.

Many mathematicians, including Conway, have seen the monster as a beautiful and still mysterious object.^[29] 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."^[30]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."^[31]

• 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

• Borcherds, R.E. (October 2002x)."What is ... the Monster?"(PDF).Notices of the American Mathematical Society.1077.
• "Monster group".MathWorld – via mathworld.wolfram.com.
• Abbott, Rachel; Bray, John N.; Linton, Steve; Nickerson, Simon J.; Norton, Simon; Parker, Richard A.; et al., eds. (17 April 2024) [4 May 1999]."Monster group". Atlas of Finite Group Representations. School of Mathematical Sciences.Queen Mary University of London. version 2.0.

• Moonshine theory
• Sporadic groups

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