Bimonster group information

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

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

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 ${\displaystyle \mathbb {Z} ^{6}:O_{5}(3):2}$, which can be reduced to the finite group ${\displaystyle 3^{5}:O_{5}(3):2}$ 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.