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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学科分类号：
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
相关论文
共 49 条
[11]  
Guralnick R.M.(1969)The non-isolated vertices in the generating graph of direct powers of simple groups Math. Z. 16 497-512
[12]  
Lucchini A.(1970)Character table and blocks of the finite simple triality groups J. Fac. Sci. Univ. Tokyo 21 93-161
[13]  
Maróti A.(1993)( Commun. Algebra 22 2221-2303
[14]  
Nagy G.P.(1994)) Commun. Algebra 234 743-792
[15]  
Britnell J.R.(2000)The probability of generating the symmetric group J. Algebra 7 134-151
[16]  
Evseev A.(1936)The conjugacy classes of Chevalley groups of type ( Q. J. Math. 36 67-87
[17]  
Guralnick R.M.(1990)) over finite fields of characteristic 2 or 3 Geom. Dedic. 56 103-113
[18]  
Holmes P.E.(1995)The semisimple conjugacy classes of finite groups of Lie type Geom. Dedic. 184 31-57
[19]  
Maróti A.(1996) and J. Algebra 137 3207-3217
[20]  
Burness T.C.(2009)The semisimple conjugacy classes and the generic class number of the finite simple groups of Lie type Proc. Am. Math. Soc. 21 133-159
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