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Bimonster group

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In mathematics, the bimonster is a group that is the wreath product of the monster group M with Z2:

The Bimonster is also a quotient of the Coxeter group corresponding to the Dynkin diagram Y555, a Y-shaped graph with 16 nodes:

Actually, the 3 outermost nodes are redundant. This is because the subgroup Y124 is the E8 Coxeter group. It generates the remaining node of Y125. This pattern extends all the way to Y444: it automatically generates the 3 extra nodes of Y555.

John H. Conway conjectured that a presentation of the bimonster could be given by adding a certain extra relation to the presentation defined by the Y444 diagram. More specifically, the affine E6 Coxeter group is , which can be reduced to the finite group by adding a single relation called the spider relation. Once this relation is added, and the diagram is extended to Y444, 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

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Many subgroups of the (bi)monster can be defined by adjoining the spider relation to smaller Coxeter diagrams, most notably the Fischer groups and the baby monster group. The groups Yij0, Yij1, Y122, Y123, and Y124 are finite even without adjoining additional relations. They are the Coxeter groups Ai+j+1, Di+j, E6, E7, and E8, respectively. Other groups, which would be infinite without the spider relation, are summarized below:

Y-group name Group generated
Y222
Y223
Y224 [note 1]
Y133 [note 2]
Y134 [note 2]
Y144 [note 2]
Y233
Y234
Y244
Y333
Y334
Y344
Y444 [note 3]
  1. ^ This is the group obtained when realizing Y224 as a subgroup of larger Y-group. However, if we simply adjoin the spider relation to the Coxeter group, we obtain the double cover .
  2. ^ a b c The spider relation can only be defined directly if the diagram has at least 2 nodes in all 3 directions. However, it is possible to define the spider relation for a larger group, then consider the subgroup generated by fewer nodes.
  3. ^ As mentioned before, the 3 outermost nodes of Y555 are redundant, so Y444 is sufficient to generate the bimonster.

See also

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References

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  • Basak, Tathagata (2007), "The complex Lorentzian Leech lattice and the Bimonster", Journal of Algebra, 309 (1): 32–56, arXiv:math/0508228, doi:10.1016/j.jalgebra.2006.05.033, MR 2301231, S2CID 125231322.
  • Ivanov, A. A. (1999), "Y-groups via Transitive Extension", Journal of Algebra, 218 (1): 142–435, doi:10.1006/jabr.1999.7882.
  • Conway, John H.; Norton, Simon P.; Soicher, Leonard H. (1988), "The Bimonster, the group Y555, and the projective plane of order 3", Computers in Algebra (Chicago, IL, 1985), Lecture Notes in Pure and Applied Mathematics, vol. 111, New York: Dekker, pp. 27–50, MR 1060755.
  • Conway, J. H.; Pritchard, A. D. (1992), "Hyperbolic reflections for the Bimonster and 3Fi24", Groups, Combinatorics & Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser., vol. 165, Cambridge: Cambridge University Press, pp. 24–45, doi:10.1017/CBO9780511629259.006, MR 1200248.
  • Conway, John H.; Simons, Christopher S. (2001), "26 implies the Bimonster", Journal of Algebra, 235 (2): 805–814, doi:10.1006/jabr.2000.8494, MR 1805481.
  • Simons, Christopher Smyth (1997), Hyperbolic reflection groups, completely replicable functions, the Monster and the Bimonster, Ph.D. thesis, Princeton University, Department of Mathematics, ISBN 978-0591-50546-7, MR 2696217.
  • Soicher, Leonard H. (1989), "From the Monster to the Bimonster", Journal of Algebra, 121 (2): 275–280, doi:10.1016/0021-8693(89)90064-1, MR 0992763.
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