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Abstract
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 language | English |
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Pages (from-to) | 1921-1927 |
Number of pages | 7 |
Journal | Proceedings of the American Mathematical Society |
Volume | 147 |
Issue number | 5 |
Early online date | 18 Jan 2019 |
DOIs | |
Publication status | Published - 31 May 2019 |
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics
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Dive into the research topics of 'Left 3-Engel elements of odd order in groups'. Together they form a unique fingerprint.Projects
- 1 Finished
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Left 3-Engel Elements in Groups
Traustason, G. (PI)
Engineering and Physical Sciences Research Council
1/08/17 → 31/05/22
Project: Research council
Profiles
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Gunnar Traustason
- Department of Mathematical Sciences - Head of Department
Person: Research & Teaching