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

Bi=M\wr \mathbb {Z} _{2}.\,

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 $\mathbb {Z} ^{6}:O_{5}(3):2$ , 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 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

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 Y_ij0, Y_ij1, Y₁₂₂, Y₁₂₃, and Y₁₂₄ are finite even without adjoining additional relations. They are the Coxeter groups A_i+j+1, D_i+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	$3^{5}:O_{5}(3):2$
Y₂₂₃	$O_{7}(3)\times 2$
Y₂₂₄	$O_{8}^{+}(3):2$ ^{[note 1]}
Y₁₃₃	$2^{7}:(O_{7}(2)\times 2)$ ^{[note 2]}
Y₁₃₄	$O_{9}(2)\times 2$ ^{[note 2]}
Y₁₄₄	$O_{10}^{-}(2):2$ ^{[note 2]}
Y₂₃₃	$2\times 2Fi_{22}$
Y₂₃₄	$2\times Fi_{23}$
Y₂₄₄	$3.Fi_{24}$
Y₃₃₃	$2\times 2^{2}.^{2}E_{6}(2)$
Y₃₃₄	$2\times 2.\mathbb {B}$
Y₃₄₄	$2\times \mathbb {M}$
Y₄₄₄	$\mathbb {M} \wr 2$ ^{[note 3]}

^ 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 $2.O_{8}^{+}(3):2$ .
^ ^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.
^ As mentioned before, the 3 outermost nodes of Y₅₅₅ are redundant, so Y₄₄₄ is sufficient to generate the bimonster.

References

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.