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(An) for n≥21 congruent to 3 modulo 18.
On the maximal number of elements pairwise generating the finite alternating group
Garonzi, Martino
Secondo
;
2024
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(An) for n≥21 congruent to 3 modulo 18.File in questo prodotto:
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