利用者:Henon/作業用ページ4

数学、特に群論において、モンスター群 （Monster group）Mとは、位数

246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 =808017424794512875886459904961710757005754368000000000 ≈8 · 10⁵³

の有限群のことを言う。この群はFischer-Griess Monsterとかthe Friendly Giantと呼ばれることもある。 この群は単純群、つまりM自身と単位群以外の正規部分群を持たない群である。 有限単純群は完全に分類されている（有限単純群の分類）。そのリストは18の可算無限系列とそれらの系列には属していない26個の散在群からなる。モンスター群はこれら散在群の中で最も大きな群であり、6個の散在群をsubquotientに含んでいる。ロバート・グライスは、この6個の散在群を"pariah"と呼び、残りの散在群を"happy family"と呼んでいる。おそらくモンスター群のもっともよい定義は、コンウェイ群とフィッシャー群をsubquotientに持つような単純群の中で最小のものという定義であろう。

存在と一意性

The Monster was predicted by Bernd Fischer (unpublished) and Robert Griess (1976) in about 1973 as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution.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. Griess (1982) constructed M as the automorphism group of the Griess algebra, a 196884-dimensional commutative nonassociative algebra. John Conway and Jacques Tits subsequently simplified this construction. Griess's construction showed that the Monster existed. John G. Thompson showed that its uniqueness (as a simple group of the given order) would follow from the existence of a 196883-dimensional faithful representation. A proof of the existence of such a representation was announced in 1982 by Simon P. Norton, though he has never published the details. The first published proof of the uniqueness of the Monster was completed by Griess, Meierfrankenfeld & Segev (1989). The character table of the Monster, a 194-by-194 array, was calculated in 1979 by Fischer and Livingstone using computer programs written by Thorne. The calculation was based on the assumption that the minimal degree of a faithful complex representation is 196883, which is the product of the 3 largest prime divisors of the order of M.

ムーンシャイン

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, 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.

マッカイによるE₈ observation

There are also connections between the monster and the extended Dynkin diagrams ${\tilde {E}}_{8},$ specifically between the nodes of the diagram and certain conjugacy classes in the monster, known as McKay's E₈ observation. This is then extended to a relation between the extended diagrams ${\tilde {E}}_{6},{\tilde {E}}_{7},{\tilde {E}}_{8}$ 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.

計算機を用いた構成

Robert A. Wilson has found explicitly (with the aid of a computer) two 196882 by 196882 matrices (with elements in the field of order 2) which together generate the Monster group; note that this is dimension 1 lower than the 196883-dimensional representation in characteristic 0. However, performing calculations with these matrices is prohibitively expensive in terms of time and storage space. Wilson with collaborators has found a method of performing calculations with the Monster that is considerably faster.

Let V be a 196882 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 3¹⁺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 gⁱu = u and gⁱv = v.

This and similar constructions (in different characteristics) have been used to prove some interesting properties of the Monster (for example, to find some of its non-local maximal subgroups).

部分群の構造

Sporadic Finite Groups Showing (Sporadic) Subgroups

The Monster has at least 43 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 is A₁₂. The Monster contains many but not all of the 26 sporadic groups as subgroups. 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 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.

そのほかの実現

モンスター群は、有理数体上のガロア群としての実現(Thompson 1984, p. 443)、フルビッツ群としての実現(Wilson 2004)が知られている。

参考文献

• J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339.
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England 1985.
• Griess, Robert L. (1976), “The structure of the monster simple group”, in Scott, W. Richard; Gross, Fletcher, Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Boston, MA: Academic Press, pp. 113–118, ISBN 978-0-12-633650-4, MR0399248 
• Griess, Robert L. (1982), “The friendly giant”, Inventiones Mathematicae 69 (1): 1–102, doi:10.1007/BF01389186, ISSN 0020-9910, MR671653 
• Griess, Robert L; Meierfrankenfeld, Ulrich; Segev, Yoav (1989), “A uniqueness proof for the Monster”, Annals of Mathematics. Second Series 130 (3): 567–602, doi:10.2307/1971455, ISSN 0003-486X, JSTOR 1971455, MR1025167, https://jstor.org/stable/1971455 
• Harada, Koichiro (2001), “Mathematics of the Monster”, Sugaku Expositions 14 (1): 55–71, ISSN 0898-9583, MR1690763 
• P. E. Holmes and R. A. Wilson, A computer construction of the Monster using 2-local subgroups, J. London Math. Soc. 67 (2003), 346–364.
• Ivanov, A. A., The Monster Group and Majorana Involutions, Cambridge tracts in mathematics, 176, Cambridge University Press, ISBN 978-0-521-88994-0 
• S. A. Linton, R. A. Parker, P. G. Walsh and R. A. Wilson, Computer construction of the Monster, J. Group Theory 1 (1998), 307–337.
• S. P. Norton, The uniqueness of the Fischer-Griess Monster, Finite groups—coming of age (Montreal, Que., 1982), 271–285, Contemp. Math., 45, Amer. Math. Soc., Providence, RI, 1985.
• M. Ronan, Symmetry and the Monster, Oxford University Press, 2006, ISBN 0-19-280722-6 (concise introduction for the lay reader).
• M. du Sautoy, Finding Moonshine, Fourth Estate, 2008, ISBN 978-0-00-721461-7 (another introduction for the lay reader; published in the US by HarperCollins as Symmetry, ISBN 978-0-06-078940-4).
• Thompson, John G. (1984), “Some finite groups which appear as Gal L/K, where K ⊆ Q(μ_n)”, Journal of Algebra 89 (2): 437–499, doi:10.1016/0021-8693(84)90228-X, MR751155 
• doi:10.1515/jgth.2001.027
  これはおそらく他の言語版からコピーされた出典です。日本語版では副テンプレートはまだ作成されていません。テンプレートページを開いて該当言語版からコピーする必要があります。通常英語版ページ

外部リンク

• MathWorld: Monster Group
• Atlas of Finite Group Representations: Monster group
• Abstruse Goose: Fischer-Griess Monster