Sporadic (Fischer-Griess) Monster group M = F1
Order = 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 = 246.320.59.76.112.133.17.19.23.29.31.41.47.59.71.
Mult = 1.
Out = 1.
Robert Wilson's ATLAS page for the Fischer-Griess Monster M is available here.
M: Length ????, 2-generator, ??-relator.
< x, y | x2 = y3 = (xy)29 = [x, y]? = (xyxyxy-1)?? = ... = 1 >
Remark: x and y are R.A.Wilson's standard generators for M.
Only the insane would try to find a presentation for M on its standard generators. It is better to go for one of the Y-diagram presentations.
M × 2: Length 400, 12-generator, 79-relator (Y443).
The Coxeter group (generated by involutions a, b1, b2, b3, c1, c2, c3, d1, d2, d3, e1 and e2) such that:
e1 -- d1 -- c1 -- b1 -- a -- b2 -- c2 -- d2 -- e2
|
b3
|
c3
|
d3
along with (ab1c1ab2c2ab3c3)10 = 1 (the Spider relation).
Remark: This group is also presentated by Y553 and Y543, where the Y443 subdiagrams generate and satisfy the above presentation.
(M × M):2: Length 452, 13-generator, 92-relator (Y444).
The Coxeter group (generated by involutions a, b1, b2, b3, c1, c2, c3, d1, d2, d3, e1, e2 and e3) such that:
e1 -- d1 -- c1 -- b1 -- a -- b2 -- c2 -- d2 -- e2
|
b3
|
c3
|
d3
|
e3
along with (ab1c1ab2c2ab3c3)10 = 1 (the Spider relation).
Remark: This group is also presentated by Y555 Y554 and Y544, where the Y444 subdiagrams generate and satisfy the above presentation.
Y-diagrams.
Must mention Conway, Norton, Ivanov and others ... who actually did the work (like conjecturing and proving the above presentations).
Warning: There is more than one definition of Ypqr in existence.
Last updated 15th September, 1997
John N. Bray