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Characteristic covering numbers of finite simple groups

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Abstract

In the past few decades there has been considerable interest in word maps on groups with emphasis on (non-abelian) finite simple groups. Various asymptotic results (holding for sufficiently large groups) have been obtained. More recently non-asymptotic results (holding for all finite simple groups) emerged, with emphasis on particular words (commutators and certain power words) which are not an identity of any finite simple group. In this paper we initiate a systematic study of all words with the above property. In particular, we show that, if \(w_1, \ldots , w_6\) are words which are not an identity of any (non-abelian) finite simple group, then \(w_1(G)w_2(G) \ldots w_6(G) = G\) for all (non-abelian) finite simple groups G. Consequently, for every word w, either \(w(G)^6 = G\) for all finite simple groups, or \(w(G)=1\) for some finite simple group. These theorems follow from more general results we obtain on characteristic collections of finite groups and their covering numbers, which are of independent interest and have additional applications.

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Michael Larsen

  2. Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, 91904, Israel

    Aner Shalev

  3. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854-8019, USA

    Pham Huu Tiep

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  1. Michael Larsen
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  2. Aner Shalev
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  3. Pham Huu Tiep
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Correspondence to Aner Shalev.

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ML was partially supported by NSF grant DMS-2001349. AS was partially supported by ISF grant 686/17 and the Vinik Chair of mathematics which he holds. PT was partially supported by NSF grants DMS-1840702 and DMS-2200850, the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at the Institute for Advanced Study (Princeton). The authors were also partially supported by BSF grants 2016072 and 2020037.

The authors are grateful to Frank Lübeck for kindly providing us with the character table of the Steinberg group \({}^3\! D_4(3)\).

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Larsen, M., Shalev, A. & Tiep, P.H. Characteristic covering numbers of finite simple groups. Math. Ann. 388, 167–189 (2024). https://doi.org/10.1007/s00208-022-02520-7

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