Misplaced Pages

#15984

33-474: B is one of the 26 sporadic groups and has the second highest order of these, with the highest order being that of the monster group . The double cover of the baby monster is the centralizer of an element of order 2 in the monster group. The outer automorphism group of B is trivial and the Schur multiplier of B has order 2. The existence of this group was suggested by Bernd Fischer in unpublished work from

66-403: 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

99-554: A 196,883-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 and Jacques Tits subsequently simplified this construction. Griess's construction showed that the monster exists. Thompson showed that its uniqueness (as

132-466: A computer-free construction using the fact that its double cover is contained in the monster group. The name "baby monster" was suggested by John Horton Conway . In characteristic 0, the 4371-dimensional representation of the baby monster does not have a nontrivial invariant algebra structure analogous to the Griess algebra , but Ryba (2007) showed that it does have such an invariant algebra structure if it

165-452: 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. The mmgroup software package has been used to find two new maximal subgroups of

198-522: 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 Fi 24 , the baby monster, and the Conway group Co 1 . The Schur multiplier and the outer automorphism group of the monster are both trivial . The minimal degree of

231-524: A method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work. 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 .2.Suz.2, where Suz is the Suzuki group . Elements of

264-424: A new maximal subgroup of the form L 2 (13) and confirmed that there are no maximal subgroups with socle L 2 (8) or L 2 (16), thus completing the classification in the literature. 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

297-444: 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 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

330-403: 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 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

363-457: 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

SECTION 10

#1732870145016

396-432: Is A 12 . 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 U 3 (4), L 2 (8), and L 2 (16). However, the latter was contradicted by Dietrich et al., who found a new maximal subgroup of the form U 3 (4). The same authors had previously found

429-571: 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 A 100 and SL 20 (2) are far larger but easy to calculate with as they have "small" permutation or linear representations. Alternating groups , such as A 100 , have permutation representations that are "small" compared to

462-514: Is reduced modulo 2. The smallest faithful matrix representation of the Baby Monster is of size 4370 over the finite field of order 2. Höhn (1996) constructed a vertex operator algebra acted on by the baby monster. Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct

495-428: 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

528-537: The Bielefeld University , where he became head of mathematical sciences. In 1970, he classified the almost-simple groups generated by 3-transpositions . In the process, he discovered three new sporadic groups , which were later called the Fischer groups . His proof that the classification was complete was known but not published in a single narrative until Aschbacher published an elementary introduction of 3-transposition groups in 1996. By loosening some of

561-555: The Monster group and published a description in 1976. Fischer went on to compute the character table for both monsters, in collaboration with Donald Livingstone and Michael Thorne. Leon and Sims first produced a construction of the baby monster in 1977, and Griess produced one for the monster in 1980. Fischer was raised in Bad Endbach . He later moved to North Rhine-Westphalia , where he died in August, 2020. This article about

594-464: The conditions on his classification, in 1973 he predicted the existence of two larger sporadic simple groups: a {3,4}-transposition group, now known as the baby monster group, and a {3,4,5,6}-transposition group now known as the monster group or the Fischer-Griess Monster . These would turn out to be the two largest sporadic groups that could exist. Robert Griess independently discovered

627-439: The early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any two elements has order at most 4. He investigated its properties and computed its character table . The first construction of the baby monster was later realized as a permutation group on 13,571,955,000 points using a computer by Jeffrey Leon and Charles Sims . Robert Griess later found

660-559: The expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For the Baby monster B or F 2 , the relevant McKay–Thompson series is T 2 A ( τ ) {\displaystyle T_{2A}(\tau )} where one can set the constant term a(0) = 104 . and η ( τ ) is the Dedekind eta function . Wilson (1999) found the 30 conjugacy classes of maximal subgroups of B which are listed in

693-442: The groups 3.Fi 24 ′ , 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

SECTION 20

#1732870145016

726-510: 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 of the Griess algebra ,

759-578: The monster and the extended Dynkin diagrams E ~ 8 {\displaystyle {\tilde {E}}_{8}} 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 E ~ 6 , E ~ 7 , E ~ 8 {\displaystyle {\tilde {E}}_{6},{\tilde {E}}_{7},{\tilde {E}}_{8}} and

792-402: 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

825-560: The monster group in his June 1980 Mathematical Games column in Scientific American . The monster was predicted by Bernd Fischer (unpublished, about 1973) 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 M was found by Griess using the Thompson order formula , and Fischer, Conway , Norton and Thompson discovered other groups as subquotients, including many of

858-658: 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

891-409: The monster group. 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 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices was possible but

924-403: 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 . 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 in 1992. In this setting,

957-480: The order of an element g of the monster by finding the smallest i > 0 such that g u = u and g v = v . This and similar constructions (in different characteristics ) 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 book Symmetry and the Monster by Mark Ronan , shows how they fit together. The lines signify inclusion, as

990-460: 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." Bernd Fischer (mathematician) Bernd Fischer (18 December 1936 – 13 August 2020) was a German mathematician . He is best known for his contributions to the classification of finite simple groups , and discovered several of the sporadic groups . He introduced 3-transposition groups and constructed

1023-487: 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). Martin Seysen has implemented

Baby monster group - Misplaced Pages Continue

1056-498: The table below. Monster group 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 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

1089-622: The three Fischer groups , predicted the existence of the baby monster and monster groups , and described and computed the character table of the baby monster. He did his PhD in 1963 at the Johann Wolfgang Goethe University of Frankfurt am Main under the direction of Reinhold Baer . Fischer went to Goethe University in Frankfurt to study mathematics under Baer in the early 60s, receiving his PhD in 1963. He later moved to

#15984