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A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  17 March 2015

MICHAEL LARSEN  and
PHAM HUU TIEP
MICHAEL LARSEN
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA; [email protected]
PHAM HUU TIEP
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA; [email protected]

Abstract

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Let $w_{1}$ and $w_{2}$ be nontrivial words in free groups $F_{n_{1}}$ and $F_{n_{2}}$, respectively. We prove that, for all sufficiently large finite nonabelian simple groups $G$, there exist subsets $C_{1}\subseteq w_{1}(G)$ and $C_{2}\subseteq w_{2}(G)$ such that $|C_{i}|=O(|G|^{1/2}\log ^{1/2}|G|)$ and $C_{1}C_{2}=G$. In particular, if $w$ is any nontrivial word and $G$ is a sufficiently large finite nonabelian simple group, then $w(G)$ contains a thin base of order $2$. This is a nonabelian analog of a result of Van Vu [‘On a refinement of Waring’s problem’, Duke Math. J. 105(1) (2000), 107–134.] for the classical Waring problem. Further results concerning thin bases of $G$ of order $2$ are established for any finite group and for any compact Lie group $G$.

MSC classification

Primary: 20C33: Representations of finite groups of Lie type
Secondary: 20D06: Simple groups
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e6
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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