A note on amalgams
The author considers the typical amalgam situation. There is a group G generated by two finite subgroups \(P_ 1\), \(P_ 2\), \(P_ 1\cap P_ 2\) is a Sylow 2-subgroup of both \(P_ 1\) and \(P_ 2\), \(C_{P_ i}(O_ 2(P_ i))\subseteq O_ 2(P_ i)\), \(i=1,2\), and no nontrivial normal subgroup of G is contained in \(P_ 1\cap P_ 2\). Then as usual one introduces the coset graph. The main question is whether certain parameters of that graph are relevant for the structure of \(P_ 1\) and \(P_ 2\). For this he introduces a parameter b (which is not the same as the parameter b usually used in this area). Using this parameter he shows that it is possible to bound the number of noncentral chief factors within \(O_ 2(P_ i)\). It seems that the result is correct, but as the author introduces new objects in his proof without saying it is not possible to follow the proof. In fact the given examples are wrong as the parameters are miscalculated. Unfortunately the author uses these wrong values in his arguments.
scientific article; zbMATH DE number 4200473