On the maximal number of elements pairwise generating the finite alternating group

Let G be the alternating 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 is an element of 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, when n varies in the family of composite numbers, sigma(G)/omega(G) tends to 1 as n -> infinity. Moreover, we explicitly calculate sigma(A(n)) for n >= 21 congruent to 3 modulo 18. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).