From the Monster to the bimonster (Q1119738)

It has been noticed that the bimonster M wr 2 (wreath-product of the monster simple group M by the cyclic group of order 2) is a homomorphic image of the Coxeter group defined by the graph that is obtained by identifying the nodes a in the three graphs \(a-b_ i-c_ i-d_ i-e_ i-f_ i\) \((i=1,2,3)\). \textit{J. H. Conway}, \textit{S. P. Norton}, \textit{L. H. Soicher} [in M. C. Tangora (ed.), Computers in Algebra, 27-50 (1988)] conjecture that the factor group Y of this Coxeter group modulo the additional relation \((ab_ 1c_ 1\cdot ab_ 2c_ 2\cdot ab_ 3c_ 3)^{10}\) is in fact isomorphic to M wr 2. It is known that the generators \(f_ 1\), \(f_ 2\), \(f_ 3\) of Y are redundent and that the subgroup \(Y_{442}\) of Y generated by the remaining generators excepting \(d_ 3\) and \(e_ 3\) is correct, i.e. is isomorphic to its image \(3Fi_{24}\) in M wr 2. The present author shows that if \(Y_{443}=<Y_{442},d_ 3>\) is correct (\(\cong M\times 2)\) then Y itself is correct (\(\cong M wr 2)\). The otherwise theoretical proof involves a computer coset enumeration of 10,880 cosets when showing that a certain subgroup Q of Y is correct \((\cong O_ 9(2) wr 2)\).