Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16125220240301Conciseness on normal subgroups and new concise words from outer commutator words717719150210.30504/jims.2024.436404.1159ENG. AFernández-AlcoberDepartment of Mathematics, University of the Basque Country UPV/EHU, Bilbao, Spain.M. PintonelloDepartment of Mathematics, University of the Basque Country UPV/EHU, Bilbao, Spain.Journal Article20240118Let $w=w(x_1,\ldots,x_r)$ 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_be3a6e8b2f04c9e1558d199b8b4d8d11.pdf