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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; it has order
The Finite group 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 . Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the happy family, and the remaining six exceptions pariah group.
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.
Griess's construction showed that the monster exists. Thompson 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 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 (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 Fi24, the baby monster, and the Conway group Co1.
The Schur multiplier and the outer automorphism group of the monster are both Trivial group.
The smallest faithful permutation representation of the monster is on
The monster can be realized as a Galois group over the , 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(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 ).
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 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, "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".
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 (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup chosen is 31+12.2.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 (such as the order of an element of the monster). Wilson has exhibited vectors and whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element of the monster by finding the smallest such that and This and similar constructions (in different characteristics) were used to find some of the non-local maximal subgroups of the monster group.
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). 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.
| + Maximal subgroups of the Monster |
| centralizer of an involution of class 2A; contains the normalizer (47:23) × 2 of a Sylow 47-subgroup |
| centralizer of an involution of class 2B |
| normalizer of a subgroup of order 3 (class 3A); contains the normalizer ((29:14) × 3).2 of a Sylow 29-subgroup |
| normalizer of a Klein 4-group of type 2A2 |
| normalizer of a Klein 4-group; contains the normalizer (23:11) × S4 of a Sylow 23-subgroup |
| normalizer of a subgroup of order 3 (class 3B) |
| normalizer of a subgroup of order 3 (class 3C); contains the normalizer (31:15) × S3 of a Sylow 31-subgroup |
| normalizer of a subgroup of order 5 (class 5A) |
| normalizer of a subgroup of order 5 (class 5B) |
| normalizer of a subgroup of order 7 (class 7A) |
| contains the normalizer ((19:9) × A5):2 of a Sylow 19-subgroup |
| contains the normalizer ((17:8) × L3(2)).2 of a Sylow 17-subgroup |
| normalizer of a subgroup of order 7 (class 7B) |
| contains the normalizer (11:5 × M12):2 of a subgroup of order 11 |
| normalizer of a subgroup of order 13 (class 13A) |
| normalizer of a subgroup of order 13 (class 13B); normalizer of a Sylow 13-subgroup |
| contains the normalizer 71:35 of a Sylow 71-subgroup |
| normalizer of a Sylow 11-subgroup. |
| 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 |
| this was accidentally omitted from some previous lists of 7-local subgroups |
| previously thought to be L2(59); normalizer of a Sylow 59-subgroup |
| 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.
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. 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."
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