Bimonster group explained

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

Bi = M \wr 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

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) x 2}
Y224	O ^{+ ( 3 ): 2} _{8} ^[1]
Y133	2 ^{7 :(O} _{7 ( 2 ) x 2 )} ^[2]
Y134	O _{9(2) x 2}
Y144	O ^{- ( 2 ): 2} _{10}
Y233	2 x 2 Fi _{22}
Y234	2 x Fi _{23}
Y244	3 .Fi _{24}
Y333	2 x 2 ^{2.} ^{2 E} ^{6 ( 2 )}
Y334	2 x 2 . B
Y344	2 x M
Y444	M \wr 2 ^[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 .O ^{+ ( 3 ): 2} _{8} .
2. 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

• Triality - simple Lie group D₄, Y₁₁₁
• Affine E_6 Y₂₂₂

References

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External links

• (Note: incorrectly named here as [3^6,6,6])