Determining hypercentral Hall subgroups in finite groups

Víctor Sotomayor

Home/Archive/arXiv

arXiv · 2403.00583

Determining hypercentral Hall subgroups in finite groups

Víctor Sotomayor

View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $G$ be a finite group, and let $π$ be a set of primes. The aim of this paper is to obtain some results concerning how much information about the $π$-structure of $G$ can be gathered from the knowledge of the lengths of conjugacy classes of its $π$-elements and of their multiplicities. Among other results, we prove that this multiset of class lengths determine whether $G$ has a hypercentral Hall $π$-subgroup.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Víctor Sotomayor. 2024-08-01. Determining hypercentral Hall subgroups in finite groups. https://arxiv.org/abs/2403.00583

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Subindices and subfactors of $\mathbb{Z}_n$ and $k$-index stability of finite groups

We study subindices, subfactors, and index stability in the cyclic group $\mathbb{Z}_n$. We prove several theorems that not only confirm a conjecture and resolve some open problems about index stability of such groups, but also provide basic tools for the characterization of finite $k$-index stable groups. As a consequence, we completely characterize all 2-element index stable subsets of $\mathbb{Z}_n$, obtain an exact closed formula for their density, and determine all $n$ for which every 2-subset is index unstable. Finally, we present some problems and a research project extending the study to 3-subsets and general $k$-subsets.

math.GR↗

arXiv2026

Virtually generating graphs of pro-$p$ groups

We study the virtually generating graph of pro-$p$ groups. We show that various classes of pro-$p$ groups have connected virtually generating graph and we bound its diameter in these cases; e.g.\ compact subgroups of analytic groups over local fields and the Nottingham group.

math.GR↗

arXiv2026

Surface subgroups of Baumslag doubles along short words

If $U$ is a minimal, diskbusting, finite list of words in a free group $F_n$ of rank $n$ such that the sum of the lengths of words in $U$ is at most $2n+4$, we prove that the natural presentation complex of the Baumslag double of $F_n$ along $U$ virtually contains a $π_1$-injective embedded closed hyperbolic surface. This verifies the Tiling Conjecture of Kim and Wilton for this type of lists of words, and in particular, implies that the corresponding Baumslag double contains a hyperbolic surface subgroup.

math.GR↗