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In the area of abstract 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 20 sporadic groups (including itself) 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.^[1]

History

The monster was predicted by Bernd Fischer (unpublished, about 1973) and Robert Griess^[2] as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution.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 whether the monster actually existed. Griess^[3] constructed M as the automorphism group of the Griess algebra, a 196,884-dimensional commutative nonassociative algebra over the real numbers; he first announced his construction in Ann Arbor 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^[4] and Jacques 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 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 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 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,^[10] and as a Hurwitz 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 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).

Moonshine

The monster group is one of two principal constituents in the monstrous moonshine conjecture by Conway and Norton,^[19] 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.

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 ${\displaystyle {\tilde {E}}_{8},}$ 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 ${\displaystyle {\tilde {E}}_{6},{\tilde {E}}_{7},{\tilde {E}}_{8}}$ and the groups 3.Fi₂₄′, 2.B, and M, where these are (3/2/1-fold central extensions) of 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 monster has 46 conjugacy classes of maximal subgroups.^[16] 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.^[26] 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 46 classes of maximal subgroups of the monster are given by the following list. 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 on the list below were incorrectly omitted from some previous lists.

See also

• Supersingular prime, the prime numbers that divide the order of the monster

Citations

Sources

Further reading

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