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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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), 3207-3217
MSC (2000): Primary 05C25, 20D10, 20P05
Published electronically: June 5, 2009
MathSciNet review: 2515391
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Abstract: The generating graph $ \Gamma(G)$ of a finite group $ G$ is the graph defined on the elements of $ G$ with an edge connecting two distinct vertices if and only if they generate $ G$. The maximum size of a complete subgraph in $ \Gamma(G)$ is denoted by $ \omega(G)$. We prove that if $ G$ is a non-cyclic finite group of Fitting height at most $ 2$ that can be generated by $ 2$ elements, then $ \omega(G) = q+1$, where $ q$ is the size of a smallest chief factor of $ G$ which has more than one complement. We also show that if $ S$ is a non-abelian finite simple group and $ G$ is the largest direct power of $ S$ that can be generated by $ 2$ elements, then $ \omega(G) \leq (1+o(1))m(S)$, where $ m(S)$ denotes the minimal index of a proper subgroup in $ S$.


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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 13-15, H-1053, Budapest, Hungary
Email: maroti@renyi.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09992-4
PII: S 0002-9939(09)09992-4
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