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

2022/06/22 by Fumagalli, Francesco, Garonzi, Martino, Gheri, Pietro
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2206.11388

Abstract

Let G be the alternating 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, when n varies in the family of composite numbers, σ(G)/ω(G) tends to 1 as n → ∞. Moreover, we explicitly calculate σ(A_n) for n ≥ 21 congruent to 3 modulo 18.

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