2012/12/03 by Cristina Acciarri, Acciarri, Cristina, Pavel Shumyatsky +1 · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary 20F10 #Secondary 20E26 #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1212.0581
openalex publication_date 2012/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G∈ X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime-power, then the word wq is indeed concise in the class of residually finite groups. Further, we show that in the case where w=γk the word wq is boundedly concise in the class of residually finite groups. It remains unknown whether the word wq is actually concise in the class of all groups.