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.
On the maximal number of elements pairwise generating the symmetric group of even degree / Fumagalli F.; Garonzi M.; Maroti A.. - In: DISCRETE MATHEMATICS. - ISSN 0012-365X. - STAMPA. - 345:(2022), pp. 0-0. [10.1016/j.disc.2021.112776]
On the maximal number of elements pairwise generating the symmetric group of even degree
Fumagalli F.;Garonzi M.
;
2022
Abstract
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.File | Dimensione | Formato | |
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