Base size and generating graphs of primitive permutation groups

Abstract

In this thesis we consider base size and properties of the generating graph for finite groups.

Let Ω = {1,...,n}, let Sₙ = Sym({1,...,n}) and let G ≤ Sₙ. A base for G is a sequence Λ = (ω₁, . . . , ωₖ) of points in Ω such that the pointwise stabilizer, G_{ω₁,...,ωₖ} , is the identity. The base size of G, denoted by b(G, Ω) or b(G), is the length of the shortest base. We say that Λ is an irredundant base if

G > G_{ω₁} > G_{ω₁,ω₂} > ··· > G_{ω₁,ω₂,...,ωₖ} = 1.

If no irredundant base is longer than Λ, then we say that Λ is a maximal irredundant base for G and denote its length by I(G). A group is called large base if it is either a product action or almost simple group, and its socle is one or more copies of the alternating group Aᵣ acting on k-sets.

Let G be a primitive subgroup of Sₙ that is not large base. We prove that any irredundant base for G has size at most 5log₂n. This bound is best possible up to a small multiplicative constant and is the first logarithmic bound on the size of an irredundant base for such groups. We show that for any constant c, there are infinitely many primitive groups with maximal irredundant base size at least c times the minimal base size. As a corollary of the first result, the relational complexity of G, denoted RC(G) (see Definition 2.2.10), is at most 5log₂n + 1. In addition the maximal size of a minimal base and the height, denoted B(G) and H(G) (see Definitions 2.2.1 and 2.2.5), are both at most 5log₂n. Furthermore, we deduce that a base for G of size at most 5log₂n can be computed in polynomial time.

The generating graph Γ(G) of a finite group G has vertex set the non-identity elements of G, with two elements connected exactly when they generate G. A coclique in a graph is an empty induced subgraph, so a coclique in Γ(G) is a subset of G such that no pair of elements generate G. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of G form a coclique in Γ(G), but this coclique need not be maximal.

Let G = Sₙ or Aₙ. We first determine when the intransitive maximal subgroups of G are maximal cocliques in Γ(G), and when they are not we find the unique maximal coclique in which they are contained. We then show that for sufficiently large n, the imprimitive maximal subgroups of G are all maximal cocliques in Γ(G).

In addition, using the result on intransitive maximal subgroups we prove that a conjecture of Cameron, Lucchini, and Roney-Dougal holds for G under certain restrictions on n. Namely we prove that two elements of G have identical sets of neighbours in Γ(G) if and only if they belong to exactly the same maximal subgroups. Finally under another set of restrictions on n we then determine precisely which maximal subgroups are maximal cocliques in Γ(G).