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 | References | Similar Articles | Additional Information

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 non-cyclic 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 non-abelian 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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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:
https://doi.org/10.1090/S0002-9939-09-09992-4

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