If G is 2-generated, let ω(G) be the maximal size of a subset S of G with the property that ⟨x, y⟩ = G whenever x, y ∈ S and x ̸= y. Since any proper subgroup of G can contain at most one element of such a set S, we have ω(G) ⩽ σ(G). For a family of primitive groups G with a unique minimal normal subgroup N isomorphic to Am n and G/N cyclic, we calculate σ(G) for n divisible by 6 and m ⩾ 2. This is a generalization of a result of E. Swartz concerning the symmetric groups, which corresponds to the case m = 1. For the above family of primitive groups G, we also prove a result concerning pairwise generation: for fixed m ⩾ 2 and n even, we calculate asymptotically the value of ω(G) when n → ∞ and show that ω(G)/σ(G) tends to 1 as n → ∞. This talk is based on a joint work with Martino Garonzi (UnB).

Informações

Título: On minimal coverings and pairwise generation of some primitive groups of wreath product type
Palestrante: Júlia Arêdes de Almeida - Universidade de Brasília (UnB)
Data: 14/04/2023