Automorphism groups of primitive distance-bitransitive graphs are almost simple
Let \(\{\) A,B\(\}\) be a bipartition of a connected, nonisovalent, noncomplete bipartite graph \(\Gamma\). Let \(G\leq aut(\Gamma)\) satisfy the condition: given \(a_ 1,a_ 2\in A\) and \(b_ 1,b_ 2\in B\), with \(d(a_ 1,b_ 1)=d(a_ 2,b_ 2)\), there exists \(\phi\in G\) such that \(\phi (a_ 1)=a_ 2\) and \(\phi (b_ 1)=b_ 2\). Then (G,\(\Gamma)\) is called a distance-bitransitive pair. The main result is that, if (G,\(\Gamma)\) is such a pair, then G is almost simple, i.e., there exists a finite nonabelian simple group T such that \(T\leq G\leq aut(T)\). The proof is heavily dependent upon a theorem of \textit{C. Praeger, J. Saxl} and \textit{K. Yokoyama} [Proc. Lond. Math. Soc., III. Ser. 55, 1-21 (1987; Zbl 0619.05026)], one of the cases (concerning Hamming graphs) of which is pursued to some depth.
scientific article; zbMATH DE number 4027504