Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 5 2 2024 03 01 Conciseness on normal subgroups and new concise words from outer commutator words 71 77 191502 10.30504/jims.2024.436404.1159 EN G. A Fernández-Alcober Department of Mathematics, University of the Basque Country UPV/EHU, Bilbao, Spain. M. Pintonello Department of Mathematics, University of the Basque Country UPV/EHU, Bilbao, Spain. Journal Article 2024 01 18 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. https://jims.ims.ir/article_191502_3adabf739a9c6890dea02c56c524c857.pdf