The diameter of the generating graph of a finite soluble group

Abstract

Let G be a finite 2-generated soluble group and suppose that 〈a1,b1〉=〈a2,b2〉=G. Then there exist c1,c2 such that 〈a1,c1〉=〈c1,c2〉=〈c2,a2〉=G. Equivalently, the subgraph Δ(G) of the generating graph of a 2-generated finite soluble group G obtained by removing the isolated vertices has diameter at most 3. We construct a 2-generated group G of order 210⋅32 for which this bound is sharp. However a stronger result holds if G′ has odd order or G′ is nilpotent: in this case there exists b∈G with 〈a1,b〉=〈a2,b〉=G

The Generating Graph of Infinite Abelian Groups

“…If G is a 2-generated finite abelian group, then Γ * (G) is connected with diameter at most 2. This can be deduced from [4,Corollary 3], however a short alternative easy proof can be also given. Indeed, we can write G = P 1 × · · · × P t as a direct product of its Sylow subgroups.…”

Connectivity of generating graphs of nilpotent groups

“…A result of Whitney [17] is that κ(Γ) λ(Γ) δ(Γ). By [13], if G is a 2-generated finite nilpotent group (or more in general if G is a 2-generated finite group and the derived subgroup of G is nilpotent) then diam(∆(G)) 2, so it follows from [11,Theorem 3.3] that λ(∆(G)) = δ(∆(G)) (∆(G) is said to be maximally edge connected). Our contribution has been to show that, in fact, in this case κ(∆(G)) = λ(∆(G)) = δ(∆(G)).…”

Graphs encoding the generating properties of a finite group

“…In [24] it is proved that if G is a 2‐generator finite soluble group, then the graph obtained from the generating graph by removing the isolated vertices has a very small diameter: indeed . Moreover, if G has the property that for every non‐trivial irreducible G ‐module V which is G ‐isomorphic to a complemented chief factor of G .…”

“…Once is known that the graphs are connected in most cases, the next step is to investigate their diameters. When G is soluble and 2‐generated, it has been recently proved [24] that the graph has diameter at most 3: this bound is best possible, but it can be improved to 2 if G satisfies the following additional property: for every non‐trivial irreducible G ‐module V which is G ‐isomorphic to a complemented chief factor of G (which is true for example if the derived subgroup of G is nilpotent or has odd order). In this paper we prove a more general result (see Theorem 3.10):…”