, Volume 38, Issue 2, pp 329-350

On the generating graph of direct powers of a simple group

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Abstract

Let S be a nonabelian finite simple group and let n be an integer such that the direct product S n is 2-generated. Let Γ(S n ) be the generating graph of S n and let Γ n (S) be the graph obtained from Γ(S n ) by removing all isolated vertices. A recent result of Crestani and Lucchini states that Γ n (S) is connected, and in this note we investigate its diameter. A deep theorem of Breuer, Guralnick and Kantor implies that diam(Γ 1(S))=2, and we define Δ(S) to be the maximal n such that diam(Γ n (S))=2. We prove that Δ(S)≥2 for all S, which is best possible since Δ(A 5)=2, and we show that Δ(S) tends to infinity as |S| tends to infinity. Explicit upper and lower bounds are established for direct powers of alternating groups.