On the maximal number of elements pairwise generating the symmetric group of even degree

Let G be the symmetric group of degree n. Let ω(G) be the maximal size of a subset S of G such that 〈x,y〉=G whenever x,y∈S and x≠y and let σ(G) be the minimal size of a family of proper subgroups of G whose union is G. We prove that both functions σ(G) and ω(G) are asymptotically equal to [Formula presented] when n is even. This, together with a result of S. Blackburn, implies that σ(G)/ω(G) tends to 1 as n→∞. Moreover, we give a lower bound of n/5 on ω(G) which is independent of the classification of finite simple groups. We also calculate, for large enough n, the clique number of the graph defined as follows: the vertices are the elements of G and two vertices x,y are connected by an edge if 〈x,y〉≥An.

Let G be the symmetric group of degree n. Let omega(G) be the maximal size of a subset S of G such that (x, y) = G whenever x, y E S and x not equal y and let sigma(G) be the minimal size of a family of proper subgroups of G whose union is G. We prove that both functions sigma(G) and omega(G) are asymptotically equal to 1/2(n/n/2) when n is even. This, together with a 2 n/2 result of S. Blackburn, implies that sigma(G)/omega(G) tends to 1 as n ->infinity. Moreover, we give a lower bound of n/5 on omega(G) which is independent of the classification of finite simple groups. We also calculate, for large enough n, the clique number of the graph defined as follows: the vertices are the elements of G and two vertices x, y are connected by an edge if (x, y) >= A(n). (C) 2021 Elsevier B.V. All rights reserved.