On the clique number of the generating graph of a finite group
Authors:
Andrea Lucchini and Attila Maróti
Journal:
Proc. Amer. Math. Soc. 137 (2009), 32073217
MSC (2000):
Primary 05C25, 20D10, 20P05
Published electronically:
June 5, 2009
MathSciNet review:
2515391
Fulltext PDF Free Access
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Abstract: The generating graph of a finite group is the graph defined on the elements of with an edge connecting two distinct vertices if and only if they generate . The maximum size of a complete subgraph in is denoted by . We prove that if is a noncyclic finite group of Fitting height at most that can be generated by elements, then , where is the size of a smallest chief factor of which has more than one complement. We also show that if is a nonabelian finite simple group and is the largest direct power of that can be generated by elements, then , where denotes the minimal index of a proper subgroup in .
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 Cameron, P. J.; Ku, C. Y. Intersecting families of permutations. European J. Combin. 24 (2003), no. 7, 881890. MR 2009400 (2004g:20003)
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 Detomi, E.; Lucchini, A. Crowns and factorization of the probabilistic zeta function of a finite group. J. Algebra 265 (2003), no. 2, 651668. MR 1987022 (2004e:20119)
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 Dye, R. H. Interrelations of symplectic and orthogonal groups in characteristic two. J. Algebra 59 (1979), no. 1, 202221. MR 541675 (81c:20028)
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 Gaschütz, W. Praefrattinigruppen. Arch. Math. (Basel) 13 (1962), 418426. MR 0146262 (26:3784)
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 Guralnick, R. M.; Kantor, W. M. Probabilistic generation of finite simple groups. Special issue in honor of Helmut Wielandt. J. Algebra 234 (2000), no. 2, 743792. MR 1800754 (2002f:20038)
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 Liebeck, M. W.; Shalev, A. Classical groups, probabilistic methods, and the generation problem. Ann. of Math. (2) 144 (1996), no. 1, 77125. MR 1405944 (97e:20106a)
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 Tomkinson, M. J. Groups as the union of proper subgroups. Math. Scand. 81 (1997), 191198. MR 1613772 (99g:20042)
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Additional Information
Andrea Lucchini
Affiliation:
Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email:
lucchini@math.unipd.it
Attila Maróti
Affiliation:
Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 1315, H1053, Budapest, Hungary
Email:
maroti@renyi.hu
DOI:
http://dx.doi.org/10.1090/S0002993909099924
PII:
S 00029939(09)099924
Received by editor(s):
July 22, 2008
Published electronically:
June 5, 2009
Additional Notes:
The research of the second author was supported by OTKA NK72523, OTKA T049841, NSF Grant DMS 0140578, and by a fellowship of the Mathematical Sciences Research Institute.
Communicated by:
Jonathan I. Hall
Article copyright:
© Copyright 2009
American Mathematical Society
