\(Y\)-groups via transitive extension
In [Lond. Math. Soc. Lect. Note Ser. 165, 46-62 (1992; Zbl 0821.20005)], the author showed that the \(Y\)-group \(Y_{555}\) with the spider relation is isomorphic to the bimonster. In this paper, he gives a different proof for this result. In fact he introduces a general method which is useful for handling all the other \(Y\)-groups, too. The basic idea is to attach to a \(Y\)-group a so-called \(Y\)-graph. Let \(Z\) be a \(Y_{pqr}\)-group, \(p\geq 2\), and \(x\) be the left terminal node of the Coxeter diagram. Let \(Z\lfloor x\rfloor\) be the group generated by all Coxeter generators of \(Z\) besides \(x\). The vertices of the graph are the cosets of \(Z\lfloor x\rfloor\) and the edges are the conjugates of \((Z\lfloor x\rfloor,Z\lfloor x\rfloor x)\). The study of this graph is the main object of the paper under review. The author shows that under certain conditions this graph is triangulable and then it is possible to determine the universal cover. In this way, he shows \(Y_{222}\cong 3^5:\Omega_5(3):2\), \(Y_{322}\cong 2\times\Omega_7(3)\), \(Y_{432}\cong 2\times Fi_{23}\), \(Y_{442}\cong 3Fi_{24}\), \(Y_{333}\cong 2\times 2^2\cdot{^2E_6(2)}\), \(Y_{433}\cong 2\times 2\cdot BM\), \(Y_{443}\cong 2\times M\) and finally \(Y_{444}\cong M\wr Z_2\). Remarkably the proofs are computerfree up to \(Y_{333}\cong 2^2\cdot{^2E_6(2)}\) which depends on a double coset enumeration due to S. A. Linton.
scientific article; zbMATH DE number 1348165
scientific article