The distinguishing number of G Sym(Ω) is the smallest size of a partition of Ω such that only the identity of G fixes all the parts of the partition. Extending earlier results of Cameron, Neumann, Saxl and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for GL(2, 3) acting on the eight non-zero vectors of F 2 3 , which has distinguishing number three.

Abstract. We prove that an infinite family of semiprimitive groups are graph-restrictive. This adds to the evidence for the validity of the PSV Conjecture and increases the minimal imprimitive degree for which this conjecture is open to 12. Our result can be seen as a generalisation of the well-known theorem of Tutte on cubic graphs. The proof uses the amalgam method, adapted to this new situation.

In this paper we investigate graphs that admit a group acting arc-transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup. This type of semiprimitive group is a generalisation of an affine group. We show that if the graph has valency coprime to six, then there is a bound on the order of the vertex stabilisers depending on the valency alone. We also prove a detailed structure theorem for the vertex stabilisers in the remaining case. This is a contribution to an ongoing project to investigate the validity of the Poto\v{c}nik-Spiga-Verret Conjecture

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A transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular. Interest in this class of groups is motivated by two sources: problems arising in universal algebra related to collapsing monoids and the graphrestrictive problem for permutation groups. Here we develop a theory of semiprimitive groups which encompasses their structure, their quotient actions and a method by which all finite semiprimitive groups are constructed. We also extend some results from the theory of primitive groups to semiprimitive groups, and conclude with open problems of a similar nature. 1 primitive group are transitive. This leads to a natural generalisation where we call a permutation group quasiprimitive if each of its non-trivial normal subgroups is transitive. Many questions about permutation groups can be reduced to questions about primitive or quasiprimitive groups and they have been the focus of much attention, for example [1,8,9,10,13,14,16,17,19,23,25,26,27,28,30,31,32].Innately transitive permutation groups were introduced by Bamberg and Praeger [3] and these are the finite permutation groups G with a transitive minimal normal subgroup N. Such groups naturally occur as overgroups of quasiprimitive groups. A permutation group G is called semiregular if each point-stabiliser G ω is trivial. It is well known that the centraliser of a transitive group is semiregular [11, Theorem 4.2A] and so a normal subgroup of an innately transitive group either contains the transitive minimal normal subgroup N, and hence is itself transitive, or intersects N trivially and hence is semiregular.A permutation group is called semiprimitive if every normal subgroup is transitive or semiregular. This notion was introduced by Bereczky and Maróti [5] and was motivated by an application to collapsing transformation monoids. Their original definition required the group to be non-regular, but here we follow Potočnik, Spiga and Verret [22] and include the regular case. The class of semiprimitive groups is much wider than the class of innately transitive groups and includes all automorphism groups of graphs that are vertextransitive and locally quasiprimitive (see Lemma 8.1) and all finite Frobenius groups [5, Lemma 2.1]. Note that every normal subgroup of a semiregular group is also semiregular, therefore semiregular groups (whose theory is rather uninteresting) could be considered as the intransitive analogue of semiprimitive groups.Potočnik, Spiga and Verret were interested in semiprimitive groups due to their work on the Weiss Conjecture and its generalisations. A finite transitive permutation group L is called graph-restrictive if there is an absolute constant c(L) such that for any locally L graph-group pair (Γ, G), the order of a vertex stabiliser in G is at most c(L) (see Section 8 for more details). The Weiss Conjecture [38] asserts that any primitive group is graphrestrictive and has been proved for many classes of primitive groups, for example all 2transitive groups are graph-restrictive [36]. Praege...