Bimonster group

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

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 , which can be reduced to the finite group 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:

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])