On the generating graph of direct powers of a simple group

被引：0
作者：
Timothy C. Burness
Eleonora Crestani
机构：
[1] University of Southampton,School of Mathematics
[2] Università degli Studi di Padova,Dipartimento di Matematica
来源：
Journal of Algebraic Combinatorics | 2013年 / 38卷
关键词：
Finite simple groups; Generating graph; Diameter; Spread;
D O I：
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摘要：
Let S be a nonabelian finite simple group and let n be an integer such that the direct product Sn is 2-generated. Let Γ(Sn) be the generating graph of Sn and let Γn(S) be the graph obtained from Γ(Sn) by removing all isolated vertices. A recent result of Crestani and Lucchini states that Γn(S) is connected, and in this note we investigate its diameter. A deep theorem of Breuer, Guralnick and Kantor implies that diam(Γ1(S))=2, and we define Δ(S) to be the maximal n such that diam(Γn(S))=2. We prove that Δ(S)≥2 for all S, which is best possible since Δ(A5)=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.
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页码：329 / 350
页数：21
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