Bimonster Group
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the bimonster is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
that is the
wreath product In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used ...
of the
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; it has order :    : = 2463205976112133171923293 ...
''M'' with Z2: :Bi = M \wr \mathbb_2. \, The Bimonster is also a quotient of the
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
corresponding to the
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
''Y''555, a Y-shaped
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
with 16 nodes: : Actually, the 3 outermost nodes are redundant. This is because the subgroup ''Y''124 is the E8 Coxeter group. It generates the remaining node of ''Y''125. This pattern extends all the way to ''Y''444: it automatically generates the 3 extra nodes of ''Y''555. John H. Conway
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
d that a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
of the bimonster could be given by adding a certain extra relation to the presentation defined by the ''Y''444 diagram. More specifically, the affine E6 Coxeter group is \mathbb^6:O_5(3):2, which can be reduced to the finite group 3^5:O_5(3):2 by adding a single relation called the ''spider relation''. Once this relation is added, and the diagram is extended to ''Y''444, the group generated is the bimonster. This was proved in 1990 by Simon P. Norton; the proof was simplified in 1999 by A. A. Ivanov.


Other Y-groups

Many subgroups of the (bi)monster can be defined by adjoining the spider relation to smaller Coxeter diagrams, most notably the
Fischer groups In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by . 3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them ...
and the baby monster group. The groups ''Y''ij0, ''Y''ij1, ''Y''122, ''Y''123, and ''Y''124 are finite even without adjoining additional relations. They are the Coxeter groups ''A''i+j+1, ''D''i+j, ''E''6, ''E''7, and ''E''8, respectively. Other groups, which would be infinite without the spider relation, are summarized below:


See also

*
Triality In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8 ...
-
simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
D4, ''Y''111 * Affine E_6 ''Y''222


References

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External links

* (''Note: incorrectly named here as 6,6,6') Group theory {{group-theory-stub