Journal of the Iranian Mathematical Society

Journal of the Iranian Mathematical Society

Conciseness on normal subgroups and new concise words from outer commutator words

Document Type : Research Article

Authors
G. A Fernández-Alcober
M. Pintonello
Department of Mathematics, University of the Basque Country UPV/EHU, Bilbao, Spain.
10.30504/jims.2024.436404.1159
Abstract
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.
Keywords
Word values
verbal subgroup
concise word
Subjects
[1] G. A. Fernández-Alcober, M. Morigi, Outer commutator words are uniformly concise, J. Lond. Math. Soc. (2) 82 (2010), no. 3, 581–595.
[2] G. A. Fernández-Alcober, M. Pintonello, Conciseness on normal subgroups and new concise words from lower central and derived words, J. Iran. Math. Soc. 4 (2023), no. 2, 189–206.
Journal of the Iranian Mathematical Society
Volume 5, Issue 2
March 2024
Pages 71-77