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LEFT 3-ENGEL ELEMENTS OF ODD ORDER IN GROUPS.
- Source :
-
Proceedings of the American Mathematical Society . May2019, Vol. 147 Issue 5, p1921-1927. 7p. - Publication Year :
- 2019
-
Abstract
- Let G be a group and let x ∈ 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 w³ = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (xn+1³w³)³ = 1. [ABSTRACT FROM AUTHOR]
- Subjects :
- *FINITE groups
*GROUPS
Subjects
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 147
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 135859367
- Full Text :
- https://doi.org/10.1090/proc/14389