Bimonster group

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

:<math>Bi = M \wr \mathbb{Z}_2. \, </math>

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

:

File:BimonsterGroup.svg

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 <math>\mathbb{Z}^6:O_5(3):2</math>, which can be reduced to the finite group <math>3^5:O_5(3):2</math> 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_0ij, Y_11i, Y₁₂₂, Y₁₂₃, and Y₁₂₄ are finite even without adjoining additional relations. They are the Coxeter groups A_i+j+1, D_i+3, E₆, E₇, and E₈, respectively. Other groups, which would be infinite without the spider relation, are summarized below:

{|class="wikitable"

!Y-group name

!Group generated

|-

|Y₂₂₂

|<math>3^5:O_5(3):2</math>

|-

|Y₂₂₃

|<math>O_7(3)\times2</math>

|-

|Y₂₂₄

|<math>O_8^+(3):2</math>

|-

|Y₁₃₃

|<math>2^7:(O_7(2)\times2)</math>

|-

|Y₁₃₄

|<math>O_9(2)\times2</math>

|}