Bimonster group

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 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: