@article {
author = {Fernández-Alcober, G. and Pintonello, M.},
title = {Conciseness on normal subgroups and new concise words from lower central and derived words},
journal = {Journal of the Iranian Mathematical Society},
volume = {4},
number = {2},
pages = {189-206},
year = {2023},
publisher = {Iranian Mathematical Society},
issn = {2717-1612},
eissn = {2717-1612},
doi = {10.30504/jims.2023.392862.1105},
abstract = {Let $w=w(x_1,\ldots,x_r)$ be a lower central word or a derived 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, thus proving a generalized version of a conjecture of Azevedo and Shumyatsky. 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.},
keywords = {Word values,verbal subgroup,concise word},
url = {https://jims.ims.ir/article_175513.html},
eprint = {https://jims.ims.ir/article_175513_272bb4ad2e9a24c7a9736661c90ab65c.pdf}
}