On groups with BFC-covered word values
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arXiv
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| Format: | Preprint |
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2024
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| author | Detomi, Eloisa Morigi, Marta Shumyatsky, Pavel |
| author_facet | Detomi, Eloisa Morigi, Marta Shumyatsky, Pavel |
| contents | For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le n}. If s is a positive integer and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S of G such that |S| \le s and all w-values of G are contained in B_n(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results.
Let w be a multilinear commutator word on k variables and let G be a group satisfying the (n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that the index [G : T] and the derived length of T are both (k,n,s)-bounded.
Let G be a group satisfying the (n,s)-covering condition with respect to the word γ_k. Then
(1) γ_{2k-1}(G) has a subgroup $T$ such that the index [γ_{2k-1}(G) : T] and |T'| are both (k,n,s)-bounded; and
(2) G has a nilpotent subgroup U such that the index [G : U] and the nilpotency class of U are both (k,n,s)-bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01420 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On groups with BFC-covered word values Detomi, Eloisa Morigi, Marta Shumyatsky, Pavel Group Theory For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le n}. If s is a positive integer and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S of G such that |S| \le s and all w-values of G are contained in B_n(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results. Let w be a multilinear commutator word on k variables and let G be a group satisfying the (n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that the index [G : T] and the derived length of T are both (k,n,s)-bounded. Let G be a group satisfying the (n,s)-covering condition with respect to the word γ_k. Then (1) γ_{2k-1}(G) has a subgroup $T$ such that the index [γ_{2k-1}(G) : T] and |T'| are both (k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that the index [G : U] and the nilpotency class of U are both (k,n,s)-bounded. |
| title | On groups with BFC-covered word values |
| topic | Group Theory |
| url | https://arxiv.org/abs/2401.01420 |