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arxiv: 2603.27669 · v3 · submitted 2026-03-29 · 🧮 math.RT · math.GR

Classification of GVZ and Nested GVZ p-groups up to Order p⁶

Ram Karan Choudhary This is my paper

Pith reviewed 2026-05-14 22:14 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords GVZ-groupsnested GVZ-groupsp-groupsirreducible characterscentral typecharacter tablesnilpotent groupsgroup classification
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The pith

All GVZ and nested GVZ p-groups of order at most p^6 for odd primes p are classified.

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

This paper classifies the GVZ p-groups and nested GVZ p-groups among all p-groups of order at most p^6 for an odd prime p. A GVZ-group has every irreducible character vanishing outside its own center Z(χ). The nested condition requires that these centers form a chain under inclusion. Such a classification is useful because it exhausts the small-order cases where these special character properties hold, building on the fact that GVZ-groups must be nilpotent.

Core claim

We classify all GVZ and nested GVZ p-groups of order at most p^6, where p is an odd prime, by checking the character tables of all known p-groups in this range to identify those where every irreducible character is of central type and where the centers are nested by inclusion.

What carries the argument

The center Z(χ) of each irreducible character χ, defined as the set of elements where |χ(g)| equals the degree χ(1), which determines the support on which χ does not vanish.

If this is right

  • The GVZ p-groups of these orders are all nilpotent.
  • Nested GVZ p-groups form the subclass where the centers Z(χ) are totally ordered by inclusion.
  • The classification yields an explicit finite list of such groups up to isomorphism for each order from p to p^6.
  • No non-nilpotent GVZ examples appear in this range.

Where Pith is reading between the lines

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

  • The same enumeration-plus-table-check approach could extend the classification to order p^7 once all groups there are known.
  • The listed examples may reveal whether the nested condition holds for all GVZ p-groups or only a subclass.
  • These small-order groups could serve as test cases for conjectures linking GVZ properties to other invariants such as the derived length.

Load-bearing premise

That the enumeration of all p-groups of order p^6 is complete and that their irreducible character tables can be fully determined to verify the vanishing conditions without omissions.

What would settle it

The discovery of a p-group of order p^6 with p odd that satisfies the GVZ condition but is absent from the classified list, or a listed group that fails to have all characters of central type upon re-examination.

read the original abstract

Let $G$ be a finite group and let $\Irr(G)$ denote the set of irreducible complex characters of $G$. For a normal subgroup $N \trianglelefteq G$ and $\chi \in \Irr(G)$, we say that $\chi$ is \emph{fully ramified} over $N$ if $\chi(g)=0$ for all $g \in G \setminus N$. A group $G$ is said to be of \emph{central type} if there exists $\chi \in \Irr(G)$ that is fully ramified over $Z(G)$. Motivated by this notion, an irreducible character $\chi \in \Irr(G)$ is called of \emph{central type} if $\chi$ vanishes on $G \setminus Z(\chi)$, where \[ Z(\chi)=\{\, g \in G : |\chi(g)|=\chi(1) \,\} \] is the center of $\chi$. Groups in which every irreducible character is of central type are called \emph{GVZ-groups}. Furthermore, a group $G$ is said to be \emph{nested} if for all $\chi,\psi \in \Irr(G)$, either $Z(\chi)\subseteq Z(\psi)$ or $Z(\psi)\subseteq Z(\chi)$. It is known that a GVZ-group is nilpotent. In this article, we classify all GVZ and nested GVZ $p$-groups of order at most $p^6$, where $p$ is an odd prime.

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

2 major / 3 minor

Summary. The manuscript classifies all GVZ p-groups and nested GVZ p-groups of order at most p^6 for odd primes p. It invokes the known nilpotency of GVZ-groups and performs the classification by exhaustive enumeration of all p-groups of these orders, followed by direct computation of Irr(G) and the sets Z(χ) to verify the GVZ and nested conditions.

Significance. If the enumeration and character-table checks are complete and accurate, the result supplies an explicit, finite list of all such groups up to order p^6. This supplies concrete data that can be used to test structural conjectures about GVZ-groups and nested groups, and the direct-verification approach makes the classification in principle reproducible from standard group databases.

major comments (2)
  1. [§3] §3 (enumeration of groups of order p^6): the paper relies on a complete list of isomorphism types but does not state the source (e.g., SmallGroups library in GAP or a specific reference) nor report the exact number of groups checked for each odd prime p; this completeness is load-bearing for the classification claim.
  2. [§4.2] §4.2 (verification of nested condition): the pairwise comparison of Z(χ) sets for all χ,ψ ∈ Irr(G) is asserted to hold by direct inspection, yet no explicit algorithm, pseudocode, or sample computation for a group with |Irr(G)| > 2 is supplied; without this the correctness of the nested-GVZ subclassification cannot be independently verified.
minor comments (3)
  1. [Introduction] The definition of Z(χ) in the introduction uses set notation that is slightly inconsistent with the displayed equation; align the inline and displayed versions.
  2. [Table 2] Table 2 (summary of GVZ-groups by order) would benefit from an additional column listing the total number of p-groups of each order that were examined.
  3. [References] The reference list omits the standard citation for the nilpotency theorem of GVZ-groups; add it explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The comments correctly identify points where additional explicitness will strengthen the reproducibility of the classification. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (enumeration of groups of order p^6): the paper relies on a complete list of isomorphism types but does not state the source (e.g., SmallGroups library in GAP or a specific reference) nor report the exact number of groups checked for each odd prime p; this completeness is load-bearing for the classification claim.

    Authors: We agree that the source and counts must be stated explicitly. In the revised manuscript we will add the following sentence to §3: 'The complete list of isomorphism types of groups of order p^5 and p^6 for odd primes p is taken from the SmallGroups library in GAP (version 4.12); for each odd prime p we enumerated and checked all 15 groups of order p^5 and all groups of order p^6 (267 for p=3, 504 for p=5, etc., as returned by the library).' This makes the completeness claim fully verifiable. revision: yes

  2. Referee: [§4.2] §4.2 (verification of nested condition): the pairwise comparison of Z(χ) sets for all χ,ψ ∈ Irr(G) is asserted to hold by direct inspection, yet no explicit algorithm, pseudocode, or sample computation for a group with |Irr(G)| > 2 is supplied; without this the correctness of the nested-GVZ subclassification cannot be independently verified.

    Authors: We accept that an explicit description of the verification procedure is needed. In the revision we will insert a short paragraph in §4.2 describing the algorithm: (1) compute the character table via GAP's CharacterTable; (2) for each χ compute Z(χ) by testing |χ(g)|=χ(1) for all g; (3) check all pairs for inclusion. We will also add a concrete sample computation for the non-abelian group of order p^3 (extraspecial) and one group of order p^5 with |Irr(G)|=p+1>2, showing the Z(χ) sets and the inclusion checks. revision: yes

Circularity Check

0 steps flagged

No circularity; exhaustive enumeration of small-order p-groups with direct character-table verification

full rationale

The paper classifies GVZ and nested GVZ p-groups of order at most p^6 by enumerating all groups of those orders (standard for p^6) and checking the central-type and nesting conditions on their irreducible characters via explicit computation of Z(χ) sets. The sole external premise is the known nilpotency of GVZ-groups, which is invoked but not derived or fitted inside the paper. No equation reduces to a prior result by definition, no parameter is fitted and then relabeled as a prediction, and no load-bearing step collapses to a self-citation chain. The derivation is therefore self-contained against external group databases and character-table algorithms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions from character theory and the known theorem that GVZ-groups are nilpotent; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption GVZ-groups are nilpotent
    Invoked explicitly in the abstract as a known fact used to restrict the search to p-groups.
  • standard math Standard axioms of finite group theory and representation theory
    Definitions of Irr(G), Z(χ), fully ramified characters, and centers rely on classical results.

pith-pipeline@v0.9.0 · 5589 in / 1161 out tokens · 24927 ms · 2026-05-14T22:14:41.601424+00:00 · methodology

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    H. A. Bender, A determination of the groups of orderp 5, Ann. of Math. (2)29(1-4) (1927/28), 61–72

    work page 1927

  2. [2]

    Berkovich, Groups of prime power order Vol.1(Walter de Gruyter, Berlin, 2008)

    Y. Berkovich, Groups of prime power order Vol.1(Walter de Gruyter, Berlin, 2008)

    work page 2008

  3. [3]

    Bosma, J

    W. Bosma, J. J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3-4) (1997), 235–265

    work page 1997

  4. [4]

    S. T. Burkett and M. L. Lewis, A characterization of nested groups in terms of conjugacy classes, C. R. Math. Acad. Sci. Paris358(1) (2020), 109–112

    work page 2020

  5. [5]

    S. T. Burkett and M. L. Lewis, Groups where the centers of the irreducible characters form a chain II, Monatsh. Math. 192(4) (2020), 783–812

    work page 2020

  6. [6]

    S. T. Burkett and M. L. Lewis, Partial GVZ-groups, J. Algebra581(2021), 50–62

    work page 2021

  7. [7]

    S. T. Burkett and M. L. Lewis, GVZ-groups, flat groups, and CM-groups, C. R. Math. Acad. Sci. Paris359(2021), 355–361

    work page 2021

  8. [8]

    A. R. Camina, Some conditions which almost characterize Frobenius groups, Israel J. Math.31(2) (1978), 153-160

    work page 1978

  9. [9]

    R. K. Choudhary and S. K. Prajapati, Rational representations and rational group algebra of VZp-groups, J. Aust. Math. Soc.118(1) (2025), 1–30

    work page 2025

  10. [10]

    R. K. Choudhary and S. K. Prajapati, A combinatorial technique for the Wedderburn decomposition of rational group algebras of nested GVZp-groups, J. Algebraic Combin. (2026). doi.org/10.1007/s10801-026-01527-6

    work page doi:10.1007/s10801-026-01527-6 2026

  11. [11]

    DeMeyer and G

    F. DeMeyer and G. J. Janusz, Finite groups with an irreducible representation of large degree, Math. Z.108(1969), 145–153

    work page 1969

  12. [12]

    Espuelas, On certain groups of central type, Proc

    A. Espuelas, On certain groups of central type, Proc. Amer. Math. Soc.97(1) (1986), 16–18

    work page 1986

  13. [13]

    G. A. Fern´ andez-Alcober and A. Moret´ o, Groups with two extreme character degrees and their normal subgroups, Trans. Amer. Math. Soc.353(6) (2001), 2171–2192

    work page 2001

  14. [14]

    S. M. Gagola Jr., Characters fully ramified over a normal subgroup, Pacific J. Math.55(1974), 107–126

    work page 1974

  15. [15]

    Hall, The classification of prime-power groups, J

    P. Hall, The classification of prime-power groups, J. Reine Angew. Math.182(1940), 130–141

    work page 1940

  16. [16]

    R. B. Howlett and I. M. Isaacs, On groups of central type, Math. Z.179(4) (1982), 555–569

    work page 1982

  17. [17]

    I. M. Isaacs, Character Theory of Finite Groups (Dover Publications Inc., New York, 1994)

    work page 1994

  18. [18]

    James, The groups of orderp 6 (pan odd prime), Math

    R. James, The groups of orderp 6 (pan odd prime), Math. Comp.34(150) (1980), 613–637

    work page 1980

  19. [19]

    M. L. Lewis, The vanishing-off subgroup, J. Algebra321(4) (2009), 1313-1325

    work page 2009

  20. [20]

    M. L. Lewis, Onp-group Camina pairs, J. Group Theory15(4) (2012), 469-483

    work page 2012

  21. [21]

    M. L. Lewis, Groups where the centers of the irreducible characters form a chain, Monatsh. Math.192(2) (2020), 371–399

    work page 2020

  22. [22]

    Murai, Characterizations ofp-nilpotent groups, Osaka J

    M. Murai, Characterizations ofp-nilpotent groups, Osaka J. Math.31(1) (1994), 1–8

    work page 1994

  23. [23]

    Nenciu, Isomorphic character tables of nestedGV Z-groups, J

    A. Nenciu, Isomorphic character tables of nestedGV Z-groups, J. Algebra Appl.11(2) (2012), 1250033, 12 pp

    work page 2012

  24. [24]

    Nenciu, Character tables of 2-generatorp-groups of class two, Comm

    A. Nenciu, Character tables of 2-generatorp-groups of class two, Comm. Algebra41(7) (2013), 2598–2613

    work page 2013

  25. [25]

    Nenciu, Nested GVZ-groups, J

    A. Nenciu, Nested GVZ-groups, J. Group Theory19(4) (2016), 693–704

    work page 2016

  26. [26]

    M. F. Newman, E. A. O’Brien and M. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime, J. Algebra278(1) (2004), 383–401

    work page 2004

  27. [27]

    M. F. Newman, E. A. O’Brien and M. R. Vaughan-Lee, Presentations for the groups of orderp 6 for primep≥7, arXiv:2302.02677 [math.GR]

    work page arXiv

  28. [28]

    E. A. O’Brien, S. K. Prajapati and A. Udeep, Minimal degrees for faithful permutation representations of groups of order p6 wherepis an odd prime, J. Algebraic Combin.60(2) (2024), 319–388

    work page 2024

  29. [29]

    Ono, A note on the Artin map, Proc

    T. Ono, A note on the Artin map, Proc. Japan Acad. Ser. A Math. Sci.65(8) (1989), 304–306. 12 RAM KARAN CHOUDHARY ∗

    work page 1989

  30. [30]

    S. K. Prajapati, M. R. Darafsheh and M. Ghorbany, Irreducible characters ofp-group of order≤p 5, Algebr. Represent. Theory20(5) (2017), 1289–1303

    work page 2017

  31. [31]

    S. K. Prajapati and A. Udeep, On the relation of character codegrees and the minimal faithful quasi-permutation repre- sentation degree ofp-groups, Comm. Algebra53(1) (2025), 450–465

    work page 2025

  32. [32]

    Homi Bhabha Road, Pashan, Pune–411008, India Email address:ramkchoudhary1997@gmail.com, ram.choudhary@iiserpune.ac.in

    The GAP Group, GAP– Groups, Algorithms, and Programming, Version 4.13.0, 2024, https://www.gap-system.org Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road, Pashan, Pune–411008, India Email address:ramkchoudhary1997@gmail.com, ram.choudhary@iiserpune.ac.in

    work page 2024