Presentations of Fischer groups. II
[Part I cf. ibid. 41, No. 3, 275-335 (1992; Zbl 0757.20004).] A conjugacy class \(D\) of involutions of a group \(G\) such that the product of any two distinct elements of \(D\) has order 2 or 3 is called a class of 3-transpositions. If \(D\) generates \(G\), then \((G,D)\) is called a Fischer pair and \(D\) is called a Fischer class of \(G\). The finite groups \(G = Fi_{22},Fi_{23},Fi_{24}\) discovered by B. Fischer are each generated by a class of 3-transpositions \(D\). This article gives, for each of these three Fischer groups \(G\), a presentation \((X,R)\) where \(X\) is contained in \(D\) and \(R\) is a set of relations on \(X\) and a subset \(R_ F\) such that \((X,R_ F)\) presents a Coxeter group. Let \(H\) denote the centralizer of an element \(d\) of \(D\) where \(D\) is the Fischer class of \(G=Fi_{24}\) (resp. \(Fi_{23},Fi_{22}\)). Then \(H\) is a central extension of \(Fi_{23}\) (resp. \(Fi_{22}\), \(PSU(6,4)\)) and \(H\) has a presentation \((X_ 0,R_ 0)\) where \(X_ 0\) is contained in the Fischer class of \(H\). The desired construction proceeds by demonstrating the existence of an element \(b\) in \(D\) such that \(X = X_ 0 \cup \{b\}\) generates \(G\) and then showing that the existing theory can be applied to a set \(R\) of relations on \(X\).
scientific article; zbMATH DE number 166875
scientific article