On the non-commuting case for \((S_ 3,L_ 3(2))\)-amalgams (Q1913949)

The paper under review is one in a series of papers by the authors and W. Lempken aiming at a classification of \((S_3,H)\)-amalgams, where \(H\) is some rank 2 Lie-type group over \(\text{GF}(2)\). Here, the authors investigate rank 2 amalgams for a group \(G=\langle P_1, P_2\rangle\) over a finite group \(B=P_1 \cap P_2 \in \text{Syl}_2 (P_i)\) \((i=1, 2)\) with \(P_1/O_2(P_1) \cong S_3\) and \(P_2/O_2(P_2) \cong L_3(2)\), \(C_{P_i} (O_2 (P_i)) \leq O_2 (P_i)\) \((i=1,2)\) and \(\text{core}_G B=1\). Let \(\Gamma\) be the coset graph associated with such an amalgam. For every vertex \(\alpha \in V(\Gamma)\) the stabilizer \(G_\alpha\) is isomorphic to \(P_1\) or \(P_2\) and a pair of vertices \((\alpha, \alpha')\) is called critical, if the distance between \(\alpha\) and \(\alpha'\) is minimal with respect to \(Z_\alpha=\langle \Omega_1 (Z(S)) \mid S \in \text{Syl}_2 (G_\alpha)\rangle \not\leq O_2(G_{\alpha'})\). The authors show that in the non-commuting case, i.e. there is a critical pair \((\alpha, \alpha')\) with \([Z_\alpha, Z_{\alpha'}] \neq 1\), it follows that \(\Omega_1 (Z(P_1 \cap P_2)) \not\triangleleft P_2\). This result enables the authors to cite the Ph. D. Thesis of \textit{G. Povse} in order to list all possibilities for the chief factor structure of \(P_1\) and \(P_2\).