Year: 2025
Type: article
Abstract: A word w is concise in a class of groups $$\mathcal {C}$$ C if, for every group G in $$\mathcal {C}$$ C , the verbal subgroup w ( G ) is finite whenever w takes only finitely many values in G . This notion can be naturally extended to first-order formulae in the language of groups. We consider this more general setting and establish conciseness for various classes of groups and formulae. We prove that all formulae are concise in the class of abelian groups and that every existential formula is concise in the class of torsion-free class-2 nilpotent groups. In addition, we construct new examples of weakly rational words, which allow us to produce a wide variety of formulae that are concise in the class of residually finite groups.
Source: Monatshefte für Mathematik
Authors: Martina Conte, J. Moritz Petschick
Institution: Bielefeld University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4415210514