View in EDS

On a pseudovariety of finite supersolvable groups.

Saved in:
Bibliographic Details
Title: On a pseudovariety of finite supersolvable groups.
Authors: Marion, Claude1,2 (AUTHOR) marion@ime.usp.br, Silva, Pedro V.1 (AUTHOR) pvsilva@fc.up.pt, Tracey, Gareth3 (AUTHOR) Gareth.Tracey@warwick.ac.uk
Source: International Journal of Algebra & Computation. Aug2026, Vol. 36 Issue 5, p501-529. 29p.
Subjects: Sylow subgroups, Finite groups, Solvable groups, Profinite groups
Abstract: We introduce the pseudovariety of finite groups U = ∨ p ∈ ℙ A b (p) ∗ A b (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, and so U has decidable membership problem. We prove that it is decidable whether or not a finitely generated subgroup of a free group is closed or dense for the pro- U topology. We consider also the pseudovariety of finite groups A b (p) ∗ A b (d) (where p is a prime and d divides p − 1). We study the pro- (A b (p) ∗ A b (d)) topology on a free group and construct the unique generator of minimum size of the pseudovariety A b (p) ∗ A b (d). Finally, we prove that the variety of groups generated by U is the variety of all metabelian groups, obtaining also results on the varieties generated by a Baumslag–Solitar group of the form B S (1 , q) for q prime. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Algebra & Computation is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
Description
Abstract:We introduce the pseudovariety of finite groups U = ∨ p ∈ ℙ A b (p) ∗ A b (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, and so U has decidable membership problem. We prove that it is decidable whether or not a finitely generated subgroup of a free group is closed or dense for the pro- U topology. We consider also the pseudovariety of finite groups A b (p) ∗ A b (d) (where p is a prime and d divides p − 1). We study the pro- (A b (p) ∗ A b (d)) topology on a free group and construct the unique generator of minimum size of the pseudovariety A b (p) ∗ A b (d). Finally, we prove that the variety of groups generated by U is the variety of all metabelian groups, obtaining also results on the varieties generated by a Baumslag–Solitar group of the form B S (1 , q) for q prime. [ABSTRACT FROM AUTHOR]
ISSN:02181967
DOI:10.1142/S0218196726500189