Abstract. We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms G and are either locally (G, s)-arc transitive for s ≥ 2 or G-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of G. Given a normal subgroup N which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of N preserves both local primitivity and local s-arc transitivity and leads us to study graphs where G acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for G in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.
Motivated by the study of several problems in algebraic graph theory, we study finite primitive permutation groups whose point stabilizers are soluble. Such primitive permutation groups are divided into three types: affine, almost simple and product action, and the product action type can be reduced to the almost simple type. This paper gives an explicit list of the soluble maximal subgroups of almost simple groups. The classification is then applied to classify edge-primitive s-arc transitive graphs with s 4, solving a problem proposed by Richard M. Weiss (1999).
Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial Ginvariant partition B with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph Γ B relative to B, the bipartite subgraph Γ[B, C] of Γ induced by adjacent blocks B, C of Γ B and a certain 1-design D(B) induced by a block B ∈ B. The present paper studies the case where the size k of the blocks of D(B) satisfies k = v − 1. In the general case, where k = v − 1 2, the setwise stabilizer G B is doubly transitive on B and G is faithful on B. We prove that D(B) contains no repeated blocks if and only if Γ B is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as Γ B for some graph Γ with these properties. We prove further that Γ[B, C] % K v−1,v−1 if and only if Γ B is (G, 3)-arc transitive.
A description is given of finite permutation groups containing a cyclic regular subgroup. It is then applied to derive a classification of arc transitive circulants, completing the work dating from 1970's. It is shown that a connected arc transitive circulant of order n is one of the following: a complete graph K n , a lexicographic productwhere is a smaller arc transitive circulant, or is a normal circulant, that is, Aut has a normal cyclic regular subgroup. The description of this class of permutation groups is also used to describe the class of rotary Cayley maps in subsequent work.
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