A classification of amalgams of type \((G_ 2(2),\Sigma_ 3)\) and \((U_ 3(3),\Sigma_ 3)\) (Q1320232)

The author considers amalgams \((G = \langle P_ 1, P_ 2 \rangle, P_ 1, P_ 2, B = P_ 1 \cap P_ 2)\) of the type specified in the title and proves the following result: \(P_ 1/ O_ 2(P_ 1) \cong G_ 2(2)\), \(O_ 2(P_ 1)\) is elementary abelian of order \(2^ 6\) or \(2^ 7\), \(O_ 2(P_ 1) / C_{O_ 2(P_ 1)}(P_ 1)\) is an irreducible \(G_ 2(2)\)-module and \(O_ 2(P_ 1) \leq O_ 2(P_ 2)\). (An example of such an amalgam is provided by the Rudvalis group \(Ru\).) As the author states, this paper is part of a programme by the author, P. Rowley and W. Lempken to classify type \((S_ 3,H)\) amalgams, where \(H\) is some rank 2 Lie-type group over \(GF(2)\). (See the paper under review for further references).