On the closure of cyclic subgroups of a free group in pro-V topologies

2023/04/20 by Claude Marion, Marion, Claude, Pedro V. Silva +3
Social Sciences · Mathematics · #Japanese History and Culture #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2304.10230

Abstract

We determine the closure of a cyclic subgroup H of a free group for the pro-\bf V topology when \bf 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 and pro-V_p topologies, where G_p and V_p denote respectively the pseudovariety of finite p-groups and the pseudovariety of finite groups having a normal Sylow p-subgroup with quotient an abelian group of exponent dividing p-1. More generally, given any nonempty set P of primes, we consider the pseudovariety G_P of all finite groups having order a product of primes in P.

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