pith. sign in

arxiv: 2112.08188 · v3 · submitted 2021-12-15 · 🧮 math.GR

Gruenberg-Kegel graphs: cut groups, rational groups and the Prime Graph Question

Andreas B\"achle , Ann Kiefer , Sugandha Maheshwary , \'Angel del R\'io This is my paper

Pith reviewed 2026-05-24 12:10 UTC · model grok-4.3

classification 🧮 math.GR
keywords Gruenberg-Kegel graphcut groupsrational groupsPrime Graph Questionintegral group ringssolvable groupsnormalized units
0
0 comments X

The pith

The Prime Graph Question for integral group rings is answered for all finite rational groups and most finite cut groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper classifies the Gruenberg-Kegel graphs of finite solvable cut groups and rational groups. The Gruenberg-Kegel graph encodes the primes that divide element orders and when products of two such primes occur as orders. By determining exactly which graphs arise, the authors show that the graph of the group equals the graph of its normalized units in the integral group ring for these classes. A reader cares because this settles a structural question about units in group rings that is related to longstanding conjectures on torsion units.

Core claim

The possible Gruenberg-Kegel graphs are completely listed for finite solvable rational groups (all but one realized) and for cut groups with small prime spectrum (all realized or restricted), leading to an affirmative answer to the Prime Graph Question in these cases.

What carries the argument

The Gruenberg-Kegel graph of a group, whose vertices are primes dividing some element order and edges connect primes p and q if the group has an element of order pq; this graph is compared between a group and the group of normalized units of its integral group ring.

If this is right

  • The Prime Graph Question holds for every finite rational group.
  • The Prime Graph Question holds for most finite cut groups.
  • Metacyclic, metabelian, supersolvable, metanilpotent and 2-Frobenius groups that are cut or rational have their possible Gruenberg-Kegel graphs fully classified.
  • Most candidate graphs for solvable cut groups are realized by explicit examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the classification can be extended to non-solvable groups, the Prime Graph Question would be settled more broadly.
  • The results provide a template for checking the Prime Graph Question in other families of groups defined by properties of their integral group rings.
  • Realizing the missing graph for rational groups would complete the picture for that class.

Load-bearing premise

The groups under consideration are finite and solvable.

What would settle it

Exhibiting a finite solvable rational group whose Gruenberg-Kegel graph is not on the provided list would disprove the classification.

Figures

Figures reproduced from arXiv: 2112.08188 by Andreas B\"achle, \'Angel del R\'io, Ann Kiefer, Sugandha Maheshwary.

Figure 1
Figure 1. Figure 1: GK-graphs of solvable cut groups with at most 3 vertices. Corollary B. Let Γ be a graph with three vertices. Then Γ is the GK-graph of a finite solvable cut group if and only if 2 − 3 is an edge of Γ and either 5 or 7 is a vertex of Γ. For the remaining case of graphs with four vertices we obtain the following result which strongly restricts the possible GK-graphs of finite solvable cut groups. Theorem C. … view at source ↗
Figure 2
Figure 2. Figure 2: GK-graphs of finite solvable cut groups with 4 vertices. We also deduce which graphs can be realized as the GK-graph of a finite solvable rational group. Theorem D. Let G be a non-trivial finite solvable rational group. Then ΓGK(G) is one of the graphs (a), (c), (d), (e), (f ), (i) or (k) in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Realization of GK-graphs by solvable cut groups [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper classifies the Gruenberg-Kegel graphs of finite solvable cut groups with |π(G)| ≤ 3 completely and restricts the possible graphs for larger spectra (realizing most of them), does likewise for finite solvable rational groups (realizing all but one), and applies the results to classify the graphs for metacyclic, metabelian, supersolvable, metanilpotent and 2-Frobenius groups in these classes. As an application it answers the Prime Graph Question for the integral group rings of all finite rational groups and most finite cut groups.

Significance. If the classifications and realizations are complete, the work supplies a detailed and largely exhaustive picture of Gruenberg-Kegel graphs inside the families of cut and rational groups, together with concrete applications to several well-studied solvable classes. The resolution of the Prime Graph Question in these cases is timely in light of recent counter-examples to the First Zassenhaus Conjecture and provides falsifiable predictions that can be checked against known groups.

major comments (1)
  1. [Abstract] Abstract and §1: the headline claim that the Prime Graph Question is answered for all finite rational groups rests on the classification carried out only for finite solvable rational groups; the manuscript contains no explicit statement that every finite rational group is solvable, nor a separate argument covering non-solvable cut or rational groups.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to clarify the scope of our claims regarding rational groups. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and §1: the headline claim that the Prime Graph Question is answered for all finite rational groups rests on the classification carried out only for finite solvable rational groups; the manuscript contains no explicit statement that every finite rational group is solvable, nor a separate argument covering non-solvable cut or rational groups.

    Authors: We agree that an explicit statement is needed. It is a known result in the literature on rational groups (Q-groups) that every finite rational group is solvable; this follows from the structure of such groups (their irreducible characters being rational-valued forces the composition factors to be solvable). We will revise the abstract and §1 to include a clear statement to this effect, together with a brief reference or citation to the relevant background on rational groups. This justifies that our classification for solvable rational groups covers all finite rational groups. For cut groups the claim is already limited to 'most' finite cut groups, consistent with our focus on the solvable case. These changes will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: classifications derived from standard definitions of cut/rational groups and Gruenberg-Kegel graphs

full rationale

The paper performs a classification of Gruenberg-Kegel graphs for finite solvable cut groups (with |π(G)| ≤ 3 or restricted lists) and finite solvable rational groups, then applies this to answer the Prime Graph Question for finite rational groups and most finite cut groups. All steps rest on the pre-existing definitions of the graph (vertices = primes dividing element orders; edges when pq-elements exist), the definitions of cut groups (trivial central units in ZG) and rational groups, and standard group-theoretic constructions for the listed subclasses (metacyclic, metabelian, etc.). No parameter is fitted to data and then renamed as a prediction, no result is defined in terms of itself, and no load-bearing step reduces to a self-citation chain whose content is unverified outside the paper. The derivation is therefore self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely within the standard axioms of finite group theory and the established definitions of Gruenberg-Kegel graphs, cut groups and rational groups; no additional free parameters, ad-hoc axioms or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms and definitions of finite groups, integral group rings, and undirected graphs on primes
    All results rest on the usual background of group theory and ring theory.

pith-pipeline@v0.9.0 · 5860 in / 1195 out tokens · 28289 ms · 2026-05-24T12:10:03.050745+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages

  1. [1]

    Abbott, J

    R. Abbott, J. Bray, S. Linton, S. Nickerson, S. Norton, R. Parker, I. Suleiman, J. Tripp, P. Walsh, and R. Wilson, Atlas of finite group representations-version 3, http://brauer.maths.qmul.ac.uk/Atlas/v3/ http://brauer.maths.qmul.ac.uk/Atlas/v3/

    work page

  2. [2]

    Akhlaghi, B

    Z. Akhlaghi, B. Khosravi, and M. Khatami, Characterization by prime graph of PGL (2,p^k) where p and k>1 are odd , Internat. J. Algebra Comput. 20 (2010), no. 7, 847--873

    work page 2010

  3. [3]

    B \"a chle, Integral group rings of solvable groups with trivial central units, Forum Math

    A. B \"a chle, Integral group rings of solvable groups with trivial central units, Forum Math. 30 (2018), no. 4, 845--855

    work page 2018

  4. [4]

    Group Theory Appl

    , 3 questions on cut groups , Adv. Group Theory Appl. 8 (2019), 157--160, https://arxiv.org/abs/2001.02637 arXiv:2001.02637 [math.GR]

    work page arXiv 2019

  5. [5]

    Bovdi, T

    V. Bovdi, T. Breuer, and A. Mar\' o ti, Finite simple groups with short G alois orbits on conjugacy classes , J. Algebra 544 (2020), 151--169

    work page 2020

  6. [6]

    T. C. Burness and E. Covato, On the prime graph of simple groups, Bull. Aust. Math. Soc. 91 (2015), no. 2, 227--240

    work page 2015

  7. [7]

    B\" a chle, M

    A. B\" a chle, M. Caicedo, E. Jespers, and S. Maheshwary, Global and local properties of finite groups with only finitely many central units in their integral group ring, J. Group Theory 24 (2021), no. 6, 1163--1188

    work page 2021

  8. [8]

    B \"a chle, G

    A. B \"a chle, G. Janssens, E. Jespers, A. Kiefer, and D. Temmerman, A dichotomy for integral group rings via higher modular groups as amalgamated products, submitted (2018), 33 pages, https://arxiv.org/abs/1811.12226 arXiv:1811.12226 [math.GR]

    work page arXiv 2018

  9. [9]

    , Abelianization and fixed point properties of units in integral group rings, Math. Nachr. (2021), 54 pages, in press, https://arxiv.org/abs/1811.12184 arXiv:1811.12184 [math.GR]

    work page arXiv 2021

  10. [10]

    B \"a chle and L

    A. B \"a chle and L. Margolis, Rational conjugacy of torsion units in integral group rings of non-solvable groups, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 4, 813--830

    work page 2017

  11. [11]

    , He LP : a GAP package for torsion units in integral group rings , J. Softw. Algebra Geom. 8 (2018), 1--9

    work page 2018

  12. [12]

    , HeLP , H ertweck- L uthar- P assi method, V ersion 3.4 , https://gap-packages.github.io/HeLP https://gap-packages.github.io/ HeLP , 2018, GAP package

    work page 2018

  13. [13]

    , An application of blocks to torsion units in group rings, Proc. Amer. Math. Soc. 147 (2019), no. 10, 4221--4231

    work page 2019

  14. [14]

    Represent

    , On the prime graph question for integral group rings of 4-primary groups II , Algebr. Represent. Theory 22 (2019), no. 2, 437--457

    work page 2019

  15. [15]

    Bakshi, S

    G.K. Bakshi, S. Maheshwary, and I.B.S. Passi, Integral group rings with all central units trivial, J. Pure Appl. Algebra 221 (2017), no. 8, 1955--1965

    work page 2017

  16. [16]

    Conway, R.T

    J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985

    work page 1985

  17. [17]

    Chillag and S

    D. Chillag and S. Dolfi, Semi-rational solvable groups, J. Group Theory 13 (2010), no. 4, 535--548

    work page 2010

  18. [18]

    Cameron and N.V

    P.J. Cameron and N.V. Maslova, Criterion of unrecognizability of a finite group by its G ruenberg-- K egel graph , preprint (2020), 14 pages, https://arxiv.org/abs/2012.01482 arXiv:2012.01482 [math.GR]

    work page arXiv 2020

  19. [19]

    Caicedo and L

    M. Caicedo and L. Margolis, Orders of units in integral group rings and blocks of defect 1, J. Lond. Math. Soc. (2) 103 (2021), no. 4, 1515--1546

    work page 2021

  20. [20]

    M. R. Darafsheh, A. Iranmanesh, and S. A. Moosavi, 2- F robenius Q -groups , Indian J. Pure Appl. Math. 40 (2009), no. 1, 29--34

    work page 2009

  21. [21]

    M. R. Darafsheh and H. Sharifi, Frobenius Q -groups , Arch. Math. (Basel) 83 (2004), no. 2, 102--105

    work page 2004

  22. [22]

    Eisele and L

    F. Eisele and L. Margolis, A counterexample to the first Z assenhaus conjecture , Adv. Math. 339 (2018), 599--641

    work page 2018

  23. [23]

    Ferraz, Simple components and central units in group algebras , J

    R.A. Ferraz, Simple components and central units in group algebras , J. Algebra 279 (2004), no. 1, 191--203

    work page 2004

  24. [24]

    Feit and G

    W. Feit and G. M. Seitz, On finite rational groups and related topics , Illinois Journal of Mathematics 33 (1989), no. 1, 103 -- 131

    work page 1989

  25. [25]

    The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.10.2 , 2019, https://www.gap-system.org

    work page 2019

  26. [26]

    A. L. Gavrilyuk, I. V. Khramtsov, A. S. Kondrat'ev, and N. V. Maslova, On realizability of a graph as the prime graph of a finite group, Sib. \`Elektron. Mat. Izv. 11 (2014), 246--257

    work page 2014

  27. [27]

    Gruber, T

    A. Gruber, T. M. Keller, Mark L. Lewis, K. Naughton, and B. Strasser, A characterization of the prime graphs of solvable groups, J. Algebra 442 (2015), 397--422

    work page 2015

  28. [28]

    Gow, Groups whose characters are rational-valued, J

    R. Gow, Groups whose characters are rational-valued, J. Algebra 40 (1976), no. 1, 280--299

    work page 1976

  29. [29]

    K. W. Gruenberg and K. W. Roggenkamp, Decomposition of the augmentation ideal and of the relation modules of a finite group, Proc. London Math. Soc. (3) 31 (1975), no. 2, 149--166

    work page 1975

  30. [30]

    Grittini, A note on cut groups of odd order, preprint (2020), 3 pages, https://arxiv.org/abs/1911.13196v4 arXiv:1911.13196v4 [math.GR]

    N. Grittini, A note on cut groups of odd order, preprint (2020), 3 pages, https://arxiv.org/abs/1911.13196v4 arXiv:1911.13196v4 [math.GR]

    work page arXiv 2020

  31. [31]

    Gorshkov and A

    I. Gorshkov and A. Staroletov, On groups having the prime graph as alternating and symmetric groups, Comm. Algebra 47 (2019), no. 9, 3905--3914

    work page 2019

  32. [32]

    M. A. Grechkoseeva and A. V. Vasil'ev, On the prime graph of a finite group with unique nonabelian composition factor, preprint (2021), 6 pages, https://arxiv.org/abs/2109.05860 arXiv:2109.05860v1 [math.GR]

    work page arXiv 2021

  33. [33]

    Hertweck, Partial augmentations and B rauer character values of torsion units in group rings , preprint (2007), 16 pages, arXiv.org/abs/0612429v2

    M. Hertweck, Partial augmentations and B rauer character values of torsion units in group rings , preprint (2007), 16 pages, arXiv.org/abs/0612429v2

    work page arXiv 2007

  34. [34]

    Algebra 36 (2008), no

    , The orders of torsion units in integral group rings of finite solvable groups, Comm. Algebra 36 (2008), no. 10, 3585--3588

    work page 2008

  35. [35]

    Higman, The units of group-rings , Proc

    G. Higman, The units of group-rings , Proc. London Math. Soc. (2) 46 (1940), 231--248

    work page 1940

  36. [36]

    London Math

    , Finite groups in which every element has prime power order, J. London Math. Soc. 32 (1957), 335--342

    work page 1957

  37. [37]

    Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006

    I.M. Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006

    work page 2006

  38. [38]

    Jespers and \'A

    E. Jespers and \'A . del R \' o, Group ring groups. Volume 1: Orders and generic constructions of units , Berlin: De Gruyter, 2016

    work page 2016

  39. [39]

    Kimmerle, On the prime graph of the unit group of integral group rings of finite groups, Groups, rings and algebras, Contemp

    W. Kimmerle, On the prime graph of the unit group of integral group rings of finite groups, Groups, rings and algebras, Contemp. Math., vol. 420, Amer. Math. Soc., Providence, RI, 2006, pp. 215--228

    work page 2006

  40. [40]

    Kimmerle and A

    W. Kimmerle and A. Konovalov, Recent advances on torsion subgroups of integral group rings, Groups S t A ndrews 2013, London Math. Soc. Lecture Note Ser., vol. 422, Cambridge Univ. Press, Cambridge, 2015, pp. 331--347

    work page 2013

  41. [41]

    , On the G ruenberg- K egel graph of integral group rings of finite groups , Internat. J. Algebra Comput. 27 (2017), no. 6, 619--631

    work page 2017

  42. [42]

    Bahman Khosravi, Behnam Khosravi, and Behrooz Khosravi, On the prime graph of PSL (2,p) where p>3 is a prime number , Acta Math. Hungar. 116 (2007), no. 4, 295--307

    work page 2007

  43. [43]

    A. S. Kondrat'ev, On prime graph components of finite simple groups, Math. USSR Sbornik 67 (1990), no. 1, 235--247

    work page 1990

  44. [44]

    I. S. Luthar and I. B. S. Passi, Zassenhaus conjecture for A_5 , Proc. Indian Acad. Sci. Math. Sci. 99 (1989), no. 1, 1--5

    work page 1989

  45. [45]

    M. S. Lucido, The diameter of the prime graph of a finite group, J. Group Theory 2 (1999), no. 2, 157--172

    work page 1999

  46. [46]

    Unione Mat

    Maria Silvia Lucido, Groups in which the prime graph is a tree, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 5 (2002), no. 1, 131--148

    work page 2002

  47. [47]

    Maheshwary, Integral G roup R ings W ith A ll C entral U nits T rivial: S olvable G roups , Indian J

    S. Maheshwary, Integral G roup R ings W ith A ll C entral U nits T rivial: S olvable G roups , Indian J. Pure Appl. Math. 49 (2018), no. 1, 169--175

    work page 2018

  48. [48]

    Margolis and \' A

    L. Margolis and \' A . del R\' o, Finite subgroups of group rings: a survey, Adv. Group Theory Appl. 8 (2019), 1--37

    work page 2019

  49. [49]

    Moret\' o , Field of values of cut groups and k -rational groups , J

    A. Moret\' o , Field of values of cut groups and k -rational groups , J. Algebra 591 (2022), 111--116

    work page 2022

  50. [50]

    N. V. Maslova and D. Pagon, On the realizability of a graph as the G ruenberg- K egel graph of a finite group , Sib. \`Elektron. Mat. Izv. 13 (2016), 89--100

    work page 2016

  51. [51]

    D. J. S. Robinson, A course in the theory of groups, Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1982

    work page 1982

  52. [52]

    Ritter and S.K

    J. Ritter and S.K. Sehgal, Integral group rings with trivial central units, Proc. Amer. Math. Soc. 108 (1990), no. 2, 327--329

    work page 1990

  53. [53]

    S. K. Sehgal, Units in integral group rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 69, Longman Scientific & Technical, Harlow, 1993

    work page 1993

  54. [54]

    Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc

    J. Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 578--581

    work page 1959

  55. [55]

    Trefethen, Non- A belian composition factors of finite groups with the CUT -property , J

    S. Trefethen, Non- A belian composition factors of finite groups with the CUT -property , J. Algebra 522 (2019), 236--242

    work page 2019

  56. [56]

    J. S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), no. 2, 487--513

    work page 1981

  57. [57]

    M. R. Zinov'eva and V. D. Mazurov, On finite groups with disconnected prime graph, Proc. Steklov Inst. Math. 283 (2013), no. suppl. 1, S139--S145

    work page 2013