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.
[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.
Fernández-Alcober, G., & Pintonello, M. (2024). Conciseness on normal subgroups and new concise words from outer commutator words. Journal of the Iranian Mathematical Society, 5(2), 71-77. doi: 10.30504/jims.2024.436404.1159
MLA
G. A Fernández-Alcober; M. Pintonello. "Conciseness on normal subgroups and new concise words from outer commutator words". Journal of the Iranian Mathematical Society, 5, 2, 2024, 71-77. doi: 10.30504/jims.2024.436404.1159
HARVARD
Fernández-Alcober, G., Pintonello, M. (2024). 'Conciseness on normal subgroups and new concise words from outer commutator words', Journal of the Iranian Mathematical Society, 5(2), pp. 71-77. doi: 10.30504/jims.2024.436404.1159
VANCOUVER
Fernández-Alcober, G., Pintonello, M. Conciseness on normal subgroups and new concise words from outer commutator words. Journal of the Iranian Mathematical Society, 2024; 5(2): 71-77. doi: 10.30504/jims.2024.436404.1159