Action of reflection groups on nilpotent groups (Q674632)

Let \(G\) be a group generated by a set \(X\) of involutions, such that \(o(xy)\in\{1,2,3\}\) for all \(x,y\in X\). The diagram \(\Gamma\) of \(X\) is the graph on \(X\) with the property that \(x,y\in X\) are joined by an edge iff \(o(xy)=3\). If \(G\) acts on a group \(M\), then \(M\) is called a \((G,X)\)-group provided that \([x,M]\leq C_M(y)\) for all \(x,y\in X\) with \(o(xy)=2\) and \(xy(u)=u^{-1}y(u)\) for all \(x,y\in X\) with \(o(xy)=3\) and \(u\in\{[x,m]: m\in M\}\). Under the assumption that \(M=[G,M]\) is a \((G,X)\)-group and \(\Gamma\) is connected and contains a subdiagram of type \(D_4\) it is shown that \(M\) is nilpotent of class \(\leq 2\) and \(G\) acts trivially on \([M,M]\). As an application a similiar result of \textit{B. Fischer} [Invent. Math. 13, 232-246 (1971; Zbl 0232.20040)] and \textit{J. I. Hall} [Math. Proc. Camb. Philos. Soc. 114, No. 2, 269-294 (1993; Zbl 0805.20025), section 7] for the normal subgroups \(N_i=[O_i(G),G]\), \(i\in\{2,3\}\), of a 3-transposition group \(G\) (i.e. \(X=X^G\)) can be deduced. For this purpose the groups \(N_i\) are shown to be \((G,X)\)-groups.