Let G ⩽ Sym ( Ω ) be a finite permutation group and recall that the base size of is minimal subset with trivial pointwise stabiliser. There an extensive literature on sizes for primitive groups, but there are very few results groups product type. In this paper, we initiate systematic study bases in setting. Our first main result determines every type form L ≀ P soluble point stabilisers, where ...

The main aim of this article is to study (v,k,λ)-symmetric designs admitting a flag-transitive and point-primitive automorphism group G whose socle is PSL(3,q). We indeed show that the only possible design satisfying these conditions is a Desarguesian projective plane PG(2,q) and G > PSL(3,q).