Permutation groups with conjugacy complete stabilizers (Q1339867)

Group-theoretic tools are developed to analyze the suborbits of transitive permutation groups with conjugacy complete stabilizers. It is then shown that, for \(p \equiv 3, 5 \pmod 8\), the primitive action of \(\text{PSL}(2,p)\) on the set of right cosets of \(A_ 4\) gives rise to nontrivial suborbits of lengths 4, 6, and 12. All suborbits of length 6 are self-paired, which is not the case for suborbits of lengths 4 and 12. A table gives the number of suborbits of each type. The action leads, by a standard construction, to vertex-transitive graphs with primitive automorphism group.