TY - JOUR

T1 - On the generating graph of direct powers of a simple group

AU - Burness, Tim C

AU - Crestani, Eleonora

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

U2 - 10.1007/s10801-012-0405-x

DO - 10.1007/s10801-012-0405-x

M3 - Article (Academic Journal)

VL - 38

SP - 329

EP - 350

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

ER -