2023/07/27 by Matteo Pintonello, Pintonello, Matteo, Pavel Shumyatsky +1
Computer Science · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2307.14939
openalex publication_date 2023/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A group-word w is called concise if the verbal subgroup w(G) is finite whenever w takes only finitely many values in a group G. It is known that there are words that are not concise. In particular, Olshanskii gave an example of such a word, which we denote by wo. The problem whether every word is concise in the class of residually finite groups remains wide open. In this note we observe that wo is concise in residually finite groups. Moreover, we show that wo is strongly concise in profinite groups, that is, wo(G) is finite whenever G is a profinite group in which wo takes less than 2ℵ0 values.