Sets of elements that pairwise generate a linear group

Abstract

Let G be any of the groups (P )GL(n, q), (P ) SL(n, q). Define a (simple) graph Γ = Γ (G) on the set of elements of G by connecting two vertices by an edge if and only if they generate G. Suppose that n is at least 12. Then the maximum size of a complete subgraph in Γ is equal to the chromatic number of Γ if n ≡ 2 (mod 4), or if n ≡ 2 (mod 4), q is odd and G = (P ) SL(n, q). This work was motivated by a question of Blackburn.

Some results and questions related to the generating graph of a finite group

“…The function σ(G) was much investigated for various non-solvable groups G (see [7], [32], [33], [20], [4]) and even for infinite groups G (see [29]). …”

On finite simple groups and Kneser graphs

“…In the proof of Theorem 1.1 the chromatic numbers χ(G) are calculated for various linear groups and Suzuki groups G. For linear groups G of large dimensions there is a formula for σ (G) (see [3]). In this paper the following is shown.…”

“…By [3], exact formulas are known for ω(G) and σ (G) provided that n ≥ 12. In general (for any n) we only have estimates.…”

“…Theorem 5.1 (Corollary 6.1 of [3]) When G = (P)SL(n, q) suppose that n ≥ 2 and (n, q) = (2, 5), (2,7), (2,9), (3,4). Then…”

“…, W be a list of all subspaces of V of odd dimensions greater than n/2 listed in an order such that V = U i ⊕ W i for all i with 1 ≤ i ≤ . (Such an ordering of subspaces is always possible for example by the second paragraph of Section 3 of [3].) Let V 1 , .…”