Graphs with projective linear stabilizers
Let \(H\) be a group generated by two isomorphic subgroups \(P_1\) and \(P_2\), such that \(\text{PSL}(n,q)\leq P_i\leq \text{P}\Gamma\text{L}(n,q)\) for some \(n\geq 3\) and \(q\) a power of some prime \(p\), and \(B= P_1\cap P_2\) is a one-point stabilizer in the natural action of \(P_i\) on the points of \(\text{PG}(n-1,q)\). The group \(H\) is called a completion of the amalgam \(A= (P_1,P_2,B)\), and is a homomorphic image of the universal completion \(Q= P_1*_B P_2\) with subgroup \(B\) amalgamated. The structure of the free amalgamated product \(Q\) is investigated, and the results are applied to 2-transitive graphs with projective linear stabilizers. In particular, graphs of girth 4 with projective linear stabilizers are characterized.
scientific article; zbMATH DE number 1251347