Finite-by-Nilpotent Groups and a Variation of the BFC-Theorem

For a group G and an element a ∈ G , let | a | k denote the cardinality of the set of commutators [ a , x 1 , ⋯ , x k ] , where x 1 , ⋯ , x k range over G . The main result of the paper states that a group G is finite-by-nilpotent if and only if there are positive integers k and n , such that | x |...

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Bibliographic Details
Published in:Mediterranean journal of mathematics 2022-10, Vol.19 (5), Article 202
Main Author: Shumyatsky, Pavel
Format: Article
Language:English
Subjects:
Commutators
Mathematics
Mathematics and Statistics
Subgroups
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Summary:For a group G and an element a ∈ G , let | a | k denote the cardinality of the set of commutators [ a , x 1 , ⋯ , x k ] , where x 1 , ⋯ , x k range over G . The main result of the paper states that a group G is finite-by-nilpotent if and only if there are positive integers k and n , such that | x | k ≤ n for every x ∈ G . More precisely, if | x | k ≤ n for every x ∈ G , then γ k + 1 ( G ) has finite ( k ,  n )-bounded order. Furthermore, in any group G , the set F C k ( G ) = { x ∈ G ; | x | k < ∞ } is a subgroup and γ k + 1 ( F C k ( G ) ) is locally normal.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-022-02140-0