Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.00023 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909156101849088 |
|---|---|
| author | Fernandez-Alcober, Gustavo A. Pintonello, Matteo |
| author_facet | Fernandez-Alcober, Gustavo A. Pintonello, Matteo |
| contents | Let $w=w(x_1,\ldots,x_r)$ be an outer commutator word. We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables. This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ are non-zero integers. Our approach is via the study of values of $w$ on normal subgroups, and in this setting we obtain the following result: if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite then also the subgroup generated by these values, i.e. $w(N_1,\ldots,N_r)$, is finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00023 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conciseness on normal subgroups and new concise words from outer commutator words Fernandez-Alcober, Gustavo A. Pintonello, Matteo Group Theory Let $w=w(x_1,\ldots,x_r)$ be an outer commutator word. We show that the word $w(u_1,\ldots,u_r)$ is concise whenever $u_1,\ldots,u_r$ are non-commutator words in disjoint sets of variables. This applies in particular to words of the form $w(x_1^{n_1},\ldots,x_r^{n_r})$, where the $n_i$ are non-zero integers. Our approach is via the study of values of $w$ on normal subgroups, and in this setting we obtain the following result: if $N_1,\ldots,N_r$ are normal subgroups of a group $G$ and the set of all values $w(g_1,\ldots,g_r)$ with $g_i\in N_i$ is finite then also the subgroup generated by these values, i.e. $w(N_1,\ldots,N_r)$, is finite. |
| title | Conciseness on normal subgroups and new concise words from outer commutator words |
| topic | Group Theory |
| url | https://arxiv.org/abs/2404.00023 |