On the generating graph of direct powers of a simple group

J Algebr Comb (2013) 38:329–350 DOI 10.1007/s10801-012-0405-x On the generating graph of direct powers of a simple group Timothy C. Burness · Eleonora Crestani Received: 8 August 2012 / Accepted: 5 October 2012 / Published online: 26 October 2012 © Springer Science+Business Media New York 2012 Abstract Let S be a nonabelian finite simple group and let n be an integer such that the direct product S n is 2-generated. Let Γ (S n ) be the generating graph of S n and let Γn (S) be the graph obtained from Γ (S n ) by removing all isolated vertices. A recent result of Crestani and Lucchini states that Γn (S) is connected, and in this note we investigate its diameter. A deep theorem of Breuer, Guralnick and Kantor implies that diam(Γ1 (S)) = 2, and we define Δ(S) to be the maximal n such that diam(Γn (S)) = 2. We prove that Δ(S) ≥ 2 for all S, which is best possible since Δ(A5 ) = 2, and we show that Δ(S) tends to infinity as |S| tends to infinity. Explicit upper and lower bounds are established for direct powers of alternating groups. Keywords Finite simple groups · Generating graph · Diameter · Spread 1 Introduction Let G be a finite group that can be generated by two elements and let Γ (G) be the generating graph of G; the vertices are the nontrivial elements of G, and two vertices are joined by an edge if and only if they generate G. This fascinating graph encodes many familiar generating properties. For example, G is said to be 32 -generated if every nontrivial element of G belongs to a generating pair; this is equivalent to the nonexistence of isolated vertices in Γ (G). More generally, G has spread at least k if for any k nontrivial elements x1 , . . . , xk ∈ G, there exists y ∈ G such that G = xi , y for T.C. Burness () · E. Crestani School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK e-mail: E. Crestani Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy e-mail: 330 J Algebr Comb (2013) 38:329–350 all i (this notion was introduced by Brenner and Wiegold [3] in the 1970s). Visibly, G has spread at least 2 if and only if Γ (G) is connected with diameter 2. Moreover, the graph-theoretic viewpoint suggests many new and natural questions. For instance, one can investigate the connectedness of Γ (G) (and subsequently its diameter), its (co-)clique and chromatic numbers, the existence of a Hamiltonian cycle in Γ (G), and so on. Let S be a nonabelian finite simple group. It is well known that S can be generated by two elements, and there is a vast literature in this area. Indeed, many stronger results have been established in recent years. For example, a theorem of Guralnick and Kantor [17] states that S is 32 -generated (confirming a conjecture of Steinberg [28]), and a more recent result of the same authors (with Breuer) reveals that S has spread at least 2 (see [5]). In particular, it follows that the generating graph Γ (S) has diameter 2. The clique number ω(S) of Γ (S) (that is, the size of the largest complete subgraph) has also been investigated by several authors. In [22], Liebeck and Shalev prove that there is an absolute constant c > 0 such that ω(S) ≥ c · m(S) for any S, where m(S) is the minimal index of a proper subgroup of S. In [1], Blackburn shows that if n is a sufficiently large even integer which is indivisible by 4 then ω(An ) = 2n−2 (and he also proves that this coincides with the chromatic number of Γ (An )); see [7, 23] for related results. Another recent result reveals that Γ (S) contains a Hamiltonian cycle if |S| is sufficiently large (see [6]). Let S n denote the direct product of n copies of S, and let δ(S) be the largest positive integer n such that S n is 2-generated. A formula of Philip Hall [18] states that δ(S) = φ2 (S) |Aut(S)| where φ2 (S) denotes the number of ordered pairs (x, y) such that S = x, y. In particular, δ(S) = P(S)|S|/|Out(S)| where P(S) is the probability that two randomly chosen elements generate S. For example, δ(A5 ) = 19. By a striking theorem of Liebeck and Shalev [21] (see also [13, 19]), P(S) tends to 1 as |S| tends to infinity, whence δ(S) also tends to infinity. Let n ≤ δ(S) be a positive integer and consider the generating graph Γ (S n ). If n ≥ 2 then this graph contains isolated vertices, so following [11] we define Γn (S) to be the graph obtained from Γ (S n ) by removing all the isolated vertices. By [11, Theorem 1.1], Γn (S) is connected, so it is natural to consider its diameter diam(Γn (S)). Clearly, if n < δ(S) then diam(Γn (S)) ≤ diam(Γn+1 (S)), and [11, Theorem 1.2] states that diam(Γn (S)) ≤ 4n − 2. In addition, we note that there are examples where the diameter of Γδ(S) (S) can be arbitrarily large. Indeed, [11, Theorem 1.3] states that if S = SL2 (2p ), where p is a prime, then diam(Γδ(S) (S)) ≥ 2p−2 − 1 if p is sufficiently large. We define     Δ(S) = max n : diam Γn (S) = 2 . (1) Note that diam(Γ1 (S)) = diam(Γ (S)) = 2 by the aforementioned theorem of Breuer, Guralnick and Kantor [5, Theorem 1.1], so Δ(S) ≥ 1. Our main result is the following: J Algebr Comb (2013) 38:329–350 331 Theorem 1 Let S be a nonabelian finite simple group. Then diam(Γ2 (S)) = 2, so Δ(S) ≥ 2. Moreover, Δ(S) tends to infinity as |S| tends to infinity. In Proposition 3.8, we show that Δ(A5 ) = 2, so the lower bound Δ(S) ≥ 2 in Theorem 1 is best possible. The next theorem provides explicit bounds on Δ(S) when S is an alternating group. Theorem 2 Let S = An be the alternating group of degree n ≥ 5. (i) If n is odd then   1 2 1 n − 3n + 2 ≤ Δ(S) ≤ n2 − 5n + 8 . 18 2 (ii) If n is even then  1 n(n − 1)(n − 2) ≤ Δ(S) ≤ n2 − 5n + 6 . 2 18(n2 /4 − 1) Remark 1 Note that the lower bound in part (ii) of Theorem 2 is linear in n. We refer the reader to Proposition 3.11 for a quadratic lower bound in the special case where n = 2p with p an odd prime. It would be interesting to know whether or not a quadratic lower bound exists for all even n. As one might expect, it appears to be much more difficult to obtain explicit bounds when S is a simple group of Lie type. However, the proof of Theorem 1 does provide the following lower bound on Δ(S). (Here r denotes the untwisted Lie rank of S, which is the rank of the ambient simple algebraic group.) Theorem 3 There exists an absolute constant c such that if S is a finite simple group of Lie type of rank r over Fq , where q = p f with p a prime, then either S = Sp2r (2), or Δ(S) ≥ cf −1 q r . Remark 2 Let us make some remarks on the statement of Theorem 3: (i) The proof of Theorem 1 shows that we can take c = 1/100 for the constant. (ii) The family of symplectic groups over the field of two elements is an anomaly. Indeed, it is well known that this family of groups has some unique generation properties. For example, this is the only infinite family of simple groups with exact spread two (the only other examples are A5 , A6 , and Ω8+ (2))—see [5, Corollary 1.3]. The proof of Proposition 5.5 shows that  (...truncated)