Let G be a finite insoluble group with soluble radical R(G). In this paper we investigate the soluble graph of G, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in G \ R(G), with x adjacent to y if they generate a soluble subgroup of G. Our main result states that this graph is always connected and its diameter, denoted δS (G), is at most 5. More precisely, we show that δS (G) 3 if G is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that δS (Sn) = 3 for all n 6. We also establish the existence of simple groups with δS (G) 4. For instance, we prove that δS (A2p+1) 4 for every Sophie Germain prime p 5, which demonstrates that our general upper bound of 5 is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.

For a finite group G, we investigate the direct graph Γ(G), whose vertices are the non-hypercentral elements of G and where there is an edge x → y if and only if [x, n y] = 1 for some n ∈ N. We prove that Γ(G) is always weakly connected and is strongly connected if G/Z ∞ (G) is neither Frobenius nor almost simple.

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Given a formation 𝔉, we consider the graph whose vertices are the elements of 𝐺 and where two vertices 𝑔, ℎ ∈ 𝐺 are adjacent if and only if ⟨𝑔, ℎ⟩ ∉ 𝔉. We are interested in the two following questions. Is the set of the isolated vertices of this graph a subgroup of 𝐺? Is the subgraph obtained by deleting the isolated vertices a connected graph?

We say that a finite group [Formula: see text] satisfies the independence property if, for every pair of distinct elements [Formula: see text] and [Formula: see text] of [Formula: see text], either [Formula: see text] is contained in a minimal generating set for [Formula: see text] or one of [Formula: see text] and [Formula: see text] is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups [Formula: see text] contain an element [Formula: see text] such that the maximal subgroups of [Formula: see text] containing [Formula: see text], but not containing the socle of [Formula: see text], are pairwise non-conjugate.