On the soluble graph of a finite group

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.