Archiv der Mathematik

, Volume 113, Issue 6, pp 565–570 | Cite as

Word maps in finite simple groups

  • William CockeEmail author
  • Meng-Che Ho


Elements of the free group define interesting maps, known as word maps, on groups. It was previously observed by Lubotzky that every subset of a finite simple group that is closed under endomorphisms occurs as the image of some word map. We improve upon this result by showing that the word in question can be chosen to be in \(v(\mathbf F _n),\) the verbal subgroup of the free group generated by the word v, provided that v is not a law on the finite simple group in question. In addition, we provide an example of a word w that witnesses the chirality of the Mathieu group \(M_{11}\). The paper concludes by demonstrating that not every subset of a group closed under endomorphisms occurs as the image of a word map.


Word maps Finite simple groups Chirality 

Mathematics Subject Classification

20D05 20F10 



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin–MadisonMadisonUSA
  2. 2.Department of MathematicsPurdue UniversityLafayetteUSA

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