Deflating infinite Coxeter groups to finite groups (Q2782398)

It is shown that the group presented by the Coxeter relations of the incidence graph \(\text{Inc }{\mathbf P}_2\) of the projective plane over \(\text{GF}(2)\) together with the additional relation that all free 8-gons generate symmetric groups \(S_8\) is isomorphic to the group \(O_8^-(2):2\). The proof is obtained by considering a group \(G\) presented by the Coxeter relations of the \(Y_{333}\)-diagram and repeatedly adjoining affine \(A_7\) extending nodes. Considering \(G\) as reflection group and identifying the nodes of the diagram with root vectors this amounts to repeatedly adjoining root vectors. It is then shown that by careful choice of these root vectors the process closes with the graph \(\text{Inc }{\mathbf P}_2\) and \(G\) permutes a set of 136 root vectors exactly as \(O_8^-(2):2\) does. The proof now follows by triviality of the Schur multiplier of \(O_8^-(2):2\).NEWLINENEWLINENEWLINEThe paper is closely related to an earlier paper by J. H. Conway and the author where a similar result had been proved for the Bimonster \(M\wr 2\) and the incidence of the projective plane over \(\text{GF}(3)\) starting from the \(Y_{555}\)-diagram.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00029].