On the soluble graph of a finite group

Abstract

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.

The Engel graph of a finite group

“…Moreover (2, 5)(3, 4) → 2 (1, 2, 3, 4, 5) and (1, 2, 3, 4, 5) → 2 (3, 5)(6, 7) so Ω contains all the 5cycles. Finally (1, 2, 3, 4, 5, 6, 7) → 2 (1, 4)(3, 7) and (2, 3, 5)(4, 7, 6) → 2 (1,2,3,4,5,6,7). So, Γ 2 (A 7 ) is strongly connected Now, let n = 6.…”

“…( The proof of Theorem 1.1 relies on the main result of [2], which states that the soluble graph of an insoluble group is connected and its diameter is at most 5.…”

“…So we may assume R(G) = 1. Let x 1 , x 2 be two distinct non-identical elements of G. If x 1 , x 2 is soluble then there exists 1 = f ∈ F ( x 1 , x 2 ) and x 1 → f, x 2 → f. Since the soluble graph of G is connected with diameter at most 5 (see [2]), we deduce that Γ(G) is weakly connected and its undirected diameter is at most 10, as claimed.…”

“…The previous theorem is no more true if the assumption Z ∞ (G) = 1 is removed. Consider for example G = GL (2,3). We have Z ∞ (G) = Z(G) ∼ = C 2 and G/Z(G) ∼ = S 4 .…”

“…In that case the prime divisors of |G| not contained in π 1 (G) are 7, 11, 19. However G contains a maximal subgroup M 1 ∼ = 2 3 : 7 : 3 and a maximal subgroup M 2 ∼ = PSL (2,11) and Γ(M 1 ), Γ(M 2 ) are strongly connected. There is also a maximal subgroup isomorphic to 19:6 and consequently, by Lemma 2.2, Γ(G) contains and edge from an element of order 6 to an element of order 19.…”