On the pseudovariety of groups U = \bigvee_{p ∈ ℙ} \bf Ab(p) ∗ \bf Ab(p-1)

2023/04/20 by Claude Marion, Marion, Claude, Pedro V. Silva +3
Computer Science · Mathematics · #20E05 #20E10 #20F10 #20F16 #Advanced Topology and Set Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2304.10522

openalex publication_date 2023/04/20 · openalex created_date 2023/04/24 · openalex updated_date 2026/07/28

Abstract

We introduce the pseudovariety of finite groups U = \bigvee_{p ∈ ℙ} \bf Ab(p) ∗ \bf Ab(p-1), where ℙ is the set of all primes. We show that U consists of all finite supersolvable groups with elementary abelian derived subgroup and abelian Sylow subgroups, being therefore decidable. We prove that it is decidable whether or not a finitely generated subgroup of a free group is closed or dense for the pro-\bf U topology. We consider also the pseudovariety of finite groups \bf Ab(p) ∗ \bf Ab(d) (where p is a prime and d divides p-1). We study the pro-(\bf Ab(p) ∗ \bf Ab(d)) topology on a free group and construct the unique generator of minimum size of the pseudovariety \bf Ab(p) ∗ \bf Ab(d). Finally, we prove that the variety of groups generated by \bf U is the variety of all metabelian groups, obtaining also results on the varieties generated by a Baumslag-Solitar group of the form BS(1,q) for q prime.

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