Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
-
Andreas Bächle
,
Sugandha Maheshwary
and
Ángel del Río
Abstract
The Gruenberg–Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices p, q are joined by an edge whenever the group has an element of order pq. It reflects interesting properties of the group. A group is said to be cut if the central units of its integral group ring are trivial. This is a rich family of groups, which contains the well-studied class of rational groups, and has received attention recently. In the first part of this paper, we give a complete classification of the Gruenberg–Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg–Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg–Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg–Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and 2-Frobenius groups for the classes of cut groups and rational groups, respectively. The prime graph question asks whether the Gruenberg–Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. The recent appearance of a counter-example for the first Zassenhaus conjecture on the torsion units of integral group rings has highlighted the relevance of this question. We answer the prime graph question for integral group rings for finite rational groups and most finite cut groups
Award Identifier / Grant number: INSPIRE/04/2017/000897
Funding source: Fundación Séneca
Award Identifier / Grant number: 19880/GERM/15
Funding source: Agencia Estatal de Investigación
Award Identifier / Grant number: PID2020-113206GB-I00
Funding statement: The work of the first author was partially supported by a postdoctoral fellowship of the FWO (Research Foundation Flanders). The second author is grateful to Onderzoeksraad Vrije Universiteit Brussel and to the Luxembourg Centre for Educational Testing. The third author gratefully acknowledges the support of IISER (Indian Institute of Science Education and Research) Mohali, India, DST (Department of Science and Technology), India (INSPIRE/04/2017/000897), and Universidad de Murcia, Murcia, Spain. The last author is partially supported by Grant 19880/GERM/15 funded by Fundación Séneca of Murcia, and Grant PID2020-113206GB-I00 funded by MCIN/AEI/10.13039/501100011033.
Acknowledgements
We are thankful to the referee for the careful reading and useful suggestions.
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Articles in the same Issue
- Frontmatter
- On the asymptotics of the shifted sums of Hecke eigenvalue squares
- On the splitting conjecture in the hybrid model for the Riemann zeta function
- Almost Dedekind domains without radical factorization
- Groups in which the centralizer of any non-central primary element is maximal
- GKM actions on cohomogeneity one manifolds
- Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
- Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
- Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
- Measures of noncompactness of interpolated polynomials
- The homotopy category of acyclic complexes of pure-projective modules
- The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
- Sharp Li–Yau inequalities for Dunkl harmonic oscillators
- A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
- Free polynilpotent groups and the Magnus property
