Diameters of graphs related to groups and base sizes of primitive groups

Abstract

In this thesis, we study three problems. First, we determine new bounds for base sizes b(G,Ω) of primitive subspace actions of finite almost simple classical groups G. Such base sizes are useful statistics in computational group theory. We show that if the underlying set Ω consists of k-dimensional subspaces of the natural module V = F_q^n for G, then b(G,Ω) ≥ ⌈n/k⌉ + c, where c ∈ {-2,-1,0,1} depends on n, q, k and the type of G. If instead Ω consists of pairs {X,Y} of subspaces of V with k:=dim(X) < dim(Y), and G is generated by PGL(n,q) and the graph automorphism of PSL(n,q), then b(G,Ω) ≤ max{⌈n/k⌉,4}.

The second part of the thesis concerns the intersection graph Δ_G of a finite simple group G. This graph has vertices the nontrivial proper subgroups of G, and its edges are the pairs of subgroups that intersect nontrivially. We prove that Δ_G has diameter at most 5, and that a diameter of 5 is achieved only by the graphs of the baby monster group and certain unitary groups of odd prime dimension. This answers a question posed by Shen.

Finally, we study the non-commuting, non-generating graph Ξ(G) of a group G, where G/Z(G) is either finite or non-simple. This graph is closely related to the hierarchy of graphs introduced by Cameron. The graph's vertices are the non-central elements of G, and its edges are the pairs {x,y} such that ⟨x, y⟩ ≠ G and xy ≠ yx. We show that if Ξ(G) has an edge, then either the graph is connected with diameter at most 5; the graph has exactly two connected components, each of diameter 2; or the graph consists of isolated vertices and a component of diameter at most 4. In this last case, either the nontrivial component has diameter 2, or G/Z(G) is a non-simple insoluble primitive group with every proper quotient cyclic.