Finite locally-quasiprimitive graphs (Q1348147)

A quasiprimitive permutation group contains no non-trivial non-transitive normal subgroups. A finite graph is said to be locally quasiprimitive with respect to a subgroup of automorphisms \(G\) if each vertex-stabilizer \(G_{\alpha}\) acts quasiprimitively on the neighbbors of \( \alpha \). An infinite family of locally-quasiprimitive but not locally-primitive graph (with respect to the full automorphism group) based on \( \text{PSL}(2,p)\)s is constructed in the paper; hence, the concept of local quasiprimitivity is shown to be weaker than the concept of local primitivity. The family of locally-quasiprimitive graphs is shown to be closed under the formation of the so-called normal quotients, induced by a normal subgroup, and each locally-quasiprimitive graph is shown to be a multicover of each of its normal quotients. This allows for the concept of minimal locally-quasiprimitive graphs -- the ``building stones'' of all graphs in this class -- which gives hope for possible classification of all locally-quasiprimitive graphs. Both the structure of these minimal graphs as well as the process of obtaining all locally-quasiprimitive graphs whose normal quotient is a given locally-quasiprimitive graph are considered. To facilitate the second task, the concept of an extender of a locally-quasiprimitive graph is introduced, and the simple \(G\)-admissible covers of such an extender are shown to correspond to locally-quasiprimitive normal multicovers of the original graph. The paper closes with some partial results on the problem of reconstructing a locally-quasiprimitive graph from its normal quotients, and with an extensive list of open problems.