A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank

Nick Gill, Laszlo Pyber, Endre Szabo

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    Abstract

    We prove that if $G$ is a finite simple group of Lie type and $S_1,\dots, S_k$ are subsets of $G$ satisfying $\prod_{i=1}^k|S_i|\geq|G|^c$ for some $c$ depending only on the rank of $G$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. This theorem generalizes an earlier theorem of the authors and Short.

    We also propose two conjectures that relate our result to one of Rodgers and Saxl pertaining to conjugacy classes in $\SL_n(q)$, as well as to the Product Decomposition Conjecture of Liebeck, Nikolov and Shalev.
    Original languageEnglish
    Pages (from-to)464-471
    Number of pages8
    JournalBulletin of the London Mathematical Society
    Volume52
    Issue number3
    DOIs
    Publication statusPublished - 21 May 2020

    Keywords

    • 20D06
    • 20D40 (primary)
    • 20E45 (secondary)

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