On the closure of cyclic subgroups of a free group in some profinite topologies
Year: 2025
Type: article
Abstract: We determine the closure of a cyclic subgroup 𝐻 of a free group for the pro-𝐕 topology when 𝐕 is an extension-closed pseudovariety of finite groups. We show that 𝐻 is always closed for the pro-nilpotent topology and compute its closure for the pro- G p \mathbf{G}_{p} and pro- V p \mathbf{V}_{p} topologies, where G p \mathbf{G}_{p} is the pseudovariety of finite 𝑝-groups and V p ⊃ G p \mathbf{V}_{p}\supset\mathbf{G}_{p} denote the pseudovarieties involved in the join decomposition of the pseudovariety of finite supersolvable groups. More generally, given any nonempty set 𝑃 of primes, we consider the stated problem for the pseudovariety G P \mathbf{G}_{P} of all finite groups having order a product of primes in 𝑃.
Source: Journal of Group Theory
Authors: Claude Marion, Pedro V. Silva, Gareth Tracey
Institutions: Universidade do Porto, Centro de Matemática, University of Warwick
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,712
OpenAlex ID: W4408920201