Groups, Geometry, and Dynamics


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Volume 7, Issue 4, 2013, pp. 867–882
DOI: 10.4171/GGD/208

Published online: 2013-11-20

On the product decomposition conjecture for finite simple groups

Nick Gill[1], László Pyber[2], Ian Short[3] and Endre Szabó[4]

(1) Open University, Milton Keynes, UK
(2) Hungarian Academy of Sciences, Budapest, Hungary
(3) Open University, Milton Keynes, UK
(4) Hungarian Academy of Sciences, Budapest, Hungary

We prove that if $G$ is a finite simple group of Lie type and $S$ is a subset of $G$ of size at least two, then $G$ is a product of at most $c\log|G|/\log|S|$ conjugates of $S$, where $c$ depends only on the Lie rank of $G$. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of bounded rank. We also obtain various related results about products of conjugates of a set within a group.

Keywords: Conjugacy, Doubling Lemma, Product Theorem, simple group, width

Gill Nick, Pyber László, Short Ian, Szabó Endre: On the product decomposition conjecture for finite simple groups. Groups Geom. Dyn. 7 (2013), 867-882. doi: 10.4171/GGD/208