Surjective word maps and Burnside’s paqb theorem

Robert M. Guralnick*, Martin W. Liebeck, E. A. O’Brien, Aner Shalev, Pham Huu Tiep

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x, y) ↦ xNyN is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x, y, z) ↦ xNyNzN is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map (x, y) ↦ xNyN that depend on the number of prime factors of the integer N.

Original languageAmerican English
Pages (from-to)589-695
Number of pages107
JournalInventiones Mathematicae
Issue number2
StatePublished - 1 Aug 2018

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© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.


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