Finite groups with small centralizers of word-values

  • Eloisa Detomi
  • Marta MorigiEmail author
  • Pavel Shumyatsky


Given a positive integer m and a group-word w, we consider a finite group G such that \(w(G) \ne 1\) and all centralizers of non-trivial w-values have order at most m. We prove that if \(w=v(x_1^{q_1},\dots ,x_k^{q_k})\), where v is a multilinear commutator word and \(q_1, \dots , q_k\) are p-powers for some prime p, then the order of G is bounded in terms of w and m only. Similar results hold when w is the nth Engel word or the word \(w=[x^n, y_1, \dots ,y_k]\) with \(k \ge 1\).


Group words Centralizers Commutators 

Mathematics Subject Classification

20F24 20E18 20F12 



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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Tullio Levi-Civita”Università di PadovaPaduaItaly
  2. 2.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  3. 3.Department of MathematicsUniversity of BrasiliaBrasilia-DFBrazil

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