Left 3-Engel elements of odd order in groups

Gunnar Traustason, Enrico Jabara, Pham Huu Tiep

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Let G be a group and let x ε G be a left 3-Engel element of odd order. We show that x is in the locally nilpotent radical of G. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel’manov. We also give some applications of our main result. In particular, for any given word w = w(x1, . . ., xn) in n variables, we show that if the variety of groups satisfying the law w 3 = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (x 3 n +1 w 3 ) 3 = 1.
Original languageEnglish
Pages (from-to)1921-1927
Number of pages7
JournalProceedings of the American Mathematical Society
Issue number5
Early online date18 Jan 2019
Publication statusPublished - 31 May 2019

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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