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The minimal base size for a $ p$-solvable linear group

Authors: Zoltán Halasi and Attila Maróti
Journal: Proc. Amer. Math. Soc. 144 (2016), 3231-3242
MSC (2010): Primary 20C20, 20C99; Secondary 20B99
Published electronically: December 21, 2015
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Abstract: Let $ V$ be a finite vector space over a finite field of order $ q$ and of characteristic $ p$. Let $ G\leq GL(V)$ be a $ p$-solvable completely reducible linear group. Then there exists a base for $ G$ on $ V$ of size at most $ 2$ unless $ q \leq 4$ in which case there exists a base of size at most $ 3$. This extends a recent result of Halasi and Podoski and generalizes a theorem of Seress. A generalization of a theorem of Pálfy and Wolf is also given.

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Additional Information

Zoltán Halasi
Affiliation: Department of Algebra and Number Theory, Eötvös University, 1117 Budapest, Pázmány Péter sétány 1/c, Hungary

Attila Maróti
Affiliation: Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany – and – MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, H-1053, Budapest, Hungary

Keywords: Finite group, linear representation, base size
Received by editor(s): May 10, 2015
Received by editor(s) in revised form: September 11, 2015
Published electronically: December 21, 2015
Additional Notes: The research of the first author leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 318202, from ERC Limits of discrete structures Grant No. 617747 and from OTKA K84233.
The research of the second author was supported by a Marie Curie International Reintegration Grant within the 7th European Community Framework Programme, by an Alexander von Humboldt Fellowship for Experienced Researchers, by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by OTKA K84233, by the MTA RAMKI Lendület Cryptography Research Group, and by the MTA Rényi Lendület Groups and Graphs Research Group.
Dedicated: Dedicated to the memory of Ákos Seress.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society