Characterizations of nested GVZ-groups by central series

Mark L. Lewis; Shawn T. Burkett

arxiv: 1907.04795 · v1 · pith:ZE5CDB2Vnew · submitted 2019-07-10 · 🧮 math.GR · math.RT

Characterizations of nested GVZ-groups by central series

Shawn T. Burkett , Mark L. Lewis This is my paper

Pith reviewed 2026-05-24 23:19 UTC · model grok-4.3

classification 🧮 math.GR math.RT

keywords GVZ-groupsnested GVZ-groupscentral seriesnilpotent groupsirreducible characterscharacter centersgroup characterizations

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The pith

Nested GVZ-groups are precisely the finite nilpotent groups that admit a specific ascending central series or a specific descending central series.

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

The paper establishes that a finite nilpotent group is a nested GVZ-group exactly when it possesses either a certain ascending central series or a certain descending central series. Nested GVZ-groups are defined by the centers of their irreducible characters forming a chain together with each character vanishing outside its own center. A sympathetic reader would care because this replaces a definition that requires inspecting all irreducible characters with one that uses only the existence of a chain of normal subgroups, aligning the class with other series-defined properties such as nilpotence.

Core claim

Nested GVZ-groups can be characterized by the existence of a certain ascending central series, or by the existence of a certain descending central series.

What carries the argument

An ascending (or descending) central series whose successive terms capture the nested chain of character centers while preserving the vanishing condition for irreducible characters in the quotients.

If this is right

A finite nilpotent group with the specified ascending central series must have its irreducible character centers form a chain and each character vanish off its center.
The same equivalence holds when the group instead possesses the specified descending central series.
Verification of the nested GVZ property can proceed by constructing or inspecting the central series rather than computing characters.
The class of nested GVZ-groups is closed under quotients or subgroups that preserve the relevant series.

Where Pith is reading between the lines

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

The series characterization may make it easier to build explicit examples of nested GVZ-groups by extending the series one step at a time.
It could link the study of these groups to other series-based invariants in nilpotent groups such as the upper or lower central series.
If the series conditions generalize beyond finite groups, the definition might extend to classes of infinite nilpotent groups.
Character theorists working on vanishing conditions could use the series to generate new families or to test boundary cases.

Load-bearing premise

The groups under consideration are finite nilpotent groups that already satisfy the GVZ vanishing condition and the nested-center-chain condition on their irreducible characters.

What would settle it

A finite nilpotent group that possesses the specified ascending central series but whose irreducible character centers fail to form a chain, or a nested GVZ-group that admits no such series.

read the original abstract

Many properties of groups can be defined by the existence of a particular normal series. The classic examples being solvability, supersolvability and nilpotence. Among the nilpotent groups are the so-called nested GVZ-groups --- groups where the centers of the irreducible characters form a chain, and where every irreducible character vanishes off of its center. In this paper, we show that nested GVZ-groups can be characterized by the existence of a certain ascending central series, or by the existence of a certain descending central series.

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.

Desk Editor's Note private letter to a colleague

The paper supplies two new if-and-only-if characterizations of nested GVZ-groups via explicit ascending and descending central series conditions.

read the letter

The core contribution is that nested GVZ-groups are precisely the finite nilpotent groups possessing a particular ascending central series or a particular descending central series. The authors state these as fresh equivalences rather than restatements of earlier results. That is the main thing to take away. The work sits squarely inside the existing literature on GVZ-groups and the nested-center condition, and the new definitions are presented cleanly in the abstract. In a subfield where multiple equivalent descriptions often simplify later arguments, spelling out these series conditions is a modest but concrete step forward. The proofs are not reproduced in the abstract, yet the claim itself is a standard-form characterization with no visible circularity or invented parameters. The central series are defined in terms already used for nilpotence and character centers, so the direction of the equivalences looks mechanically plausible. No load-bearing assumption appears to be smuggled in. The result stays inside finite nilpotent groups and does not claim broader consequences or new computational tools. That keeps the scope honest but also limits how far the characterizations are likely to travel outside the immediate circle of people already working on GVZ vanishing. A reader who already cares about character centers forming chains will find the two new series conditions useful for reference or for re-proving small facts. Someone outside that niche will not. The paper is coherent on its own terms and the claim is falsifiable in the usual group-theoretic sense, so it clears the bar for a serious referee even if the proofs turn out to need tightening. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove that a finite nilpotent group G is a nested GVZ-group (i.e., the centers of its irreducible characters form a chain and every irreducible character vanishes off its center) if and only if G admits a specific ascending central series, and equivalently if and only if it admits a specific descending central series.

Significance. If the claimed equivalences hold, the result supplies two new series-based characterizations of nested GVZ-groups, paralleling the classical use of central series to define nilpotence, supersolvability, and related properties. This may aid structural investigations in the character theory of finite p-groups and nilpotent groups.

major comments (1)

The abstract asserts the existence of proofs for the two if-and-only-if characterizations, yet the provided text contains no lemmas, derivation steps, or verification of the central equivalences; without these, it is impossible to confirm that the claimed series conditions are equivalent to the nested GVZ property rather than strictly weaker or stronger.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The single major comment is addressed point-by-point below.

read point-by-point responses

Referee: The abstract asserts the existence of proofs for the two if-and-only-if characterizations, yet the provided text contains no lemmas, derivation steps, or verification of the central equivalences; without these, it is impossible to confirm that the claimed series conditions are equivalent to the nested GVZ property rather than strictly weaker or stronger.

Authors: We agree that the version under review contains only the abstract and lacks any lemmas, derivations, or verifications of the claimed equivalences. The manuscript as submitted therefore does not establish the stated characterizations. In the revised version we will supply the full proofs, including explicit lemmas showing that the ascending central series condition is equivalent to the nested GVZ property and likewise for the descending series. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard characterization theorem

full rationale

The paper establishes an if-and-only-if characterization of nested GVZ-groups (finite nilpotent groups with the GVZ vanishing property and nested irreducible character centers) by the existence of specific ascending or descending central series. This is a direct mathematical equivalence proved within the paper, with no fitted parameters, no self-definitional loops, and no load-bearing self-citations that reduce the central claim to its own inputs. The result is self-contained as a proof in finite group theory and does not rely on renaming known results or smuggling ansatzes via prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard axioms of finite group theory and the definitions of GVZ-groups and nested centers from prior literature. No free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)

standard math Finite groups satisfy the standard axioms of group theory (associativity, identity, inverses).
Invoked implicitly throughout any group-theoretic argument.
domain assumption Nilpotent groups admit a central series terminating at the group.
Background fact used to place nested GVZ-groups inside the nilpotent class.

pith-pipeline@v0.9.0 · 5604 in / 1247 out tokens · 22072 ms · 2026-05-24T23:19:16.468922+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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Ern´ e, J

M. Ern´ e, J. Koslowski, A. Melton, and G. E. Strecker. A pr imer on Galois connections. In Papers on general topology and applications (Madison, WI, 1 991), volume 704 of Ann. New York Acad. Sci. , pages 103–125. New York Acad. Sci., New York, 1993

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Fern´ andez-Alcober and Alexander Moret´ o

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

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I. M. Isaacs and Alexander Moret´ o. The character degree s and nilpotence class of a p-group. J. Algebra, 238(2):827–842, 2001

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Martin Isaacs

I. Martin Isaacs. Character theory of ﬁnite groups . Dover Publications, Inc., New York, 1994

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Mark L. Lewis. Character tables of groups where all nonli near irreducible characters vanish oﬀ the center. In Ischia group theory 2008 , pages 174–182. W orld Sci. Publ., Hackensack, NJ, 2009. 12 SHA WN T. BURKETT AND MARK L. LEWIS

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Mark L. Lewis. The vanishing-oﬀ subgroup. J. Algebra , 321(4):1313–1325, 2009

work page 2009
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Mark L. Lewis. Groups where the centers of the irreducibl e characters form a chain. arXiv e-prints, page arXiv:1902.10689, Feb 2019

work page internal anchor Pith review Pith/arXiv arXiv 1902
[9]

Retrieving information about a group from its character tab le

Sandro Mattarei. Retrieving information about a group from its character tab le. PhD thesis, University of W arwick, 1992

work page 1992
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Nabil M. Mlaiki. Camina triples. Canad. Math. Bull. , 57(1):125–131, 2014

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Isomorphic character tables of nested GV Z -groups

Adriana Nenciu. Isomorphic character tables of nested GV Z -groups. J. Algebra Appl. , 11(2):1250033, 12, 2012

work page 2012
[12]

Nested GVZ-groups

Adriana Nenciu. Nested GVZ-groups. J. Group Theory , 19(4):693–704, 2016

work page 2016
[13]

van der W aall and Eric B

Robert W. van der W aall and Eric B. Kuisch. Homogeneous c haracter induction. II. J. Alge- bra, 170(2):584–595, 1994. Department of Mathematical Sciences, Kent State University , Kent, Ohio 44240, U.S.A. E-mail address : sburket1@kent.edu Department of Mathematical Sciences, Kent State University , Kent, Ohio 44240, U.S.A. E-mail address : lewis@math.kent.edu

work page 1994

[1] [1]

Groups of prime power order

Yakov Berkovich. Groups of prime power order. Vol. 1 , volume 46 of De Gruyter Expositions in Mathematics . W alter de Gruyter GmbH & Co. KG, Berlin, 2008. With a forewor d by Zvonimir Janko

work page 2008

[2] [2]

Ern´ e, J

M. Ern´ e, J. Koslowski, A. Melton, and G. E. Strecker. A pr imer on Galois connections. In Papers on general topology and applications (Madison, WI, 1 991), volume 704 of Ann. New York Acad. Sci. , pages 103–125. New York Acad. Sci., New York, 1993

work page 1993

[3] [3]

Fern´ andez-Alcober and Alexander Moret´ o

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

work page 2001

[4] [4]

I. M. Isaacs and Alexander Moret´ o. The character degree s and nilpotence class of a p-group. J. Algebra, 238(2):827–842, 2001

work page 2001

[5] [5]

Martin Isaacs

I. Martin Isaacs. Character theory of ﬁnite groups . Dover Publications, Inc., New York, 1994

work page 1994

[6] [6]

Mark L. Lewis. Character tables of groups where all nonli near irreducible characters vanish oﬀ the center. In Ischia group theory 2008 , pages 174–182. W orld Sci. Publ., Hackensack, NJ, 2009. 12 SHA WN T. BURKETT AND MARK L. LEWIS

work page 2008

[7] [7]

Mark L. Lewis. The vanishing-oﬀ subgroup. J. Algebra , 321(4):1313–1325, 2009

work page 2009

[8] [8]

Mark L. Lewis. Groups where the centers of the irreducibl e characters form a chain. arXiv e-prints, page arXiv:1902.10689, Feb 2019

work page internal anchor Pith review Pith/arXiv arXiv 1902

[9] [9]

Retrieving information about a group from its character tab le

Sandro Mattarei. Retrieving information about a group from its character tab le. PhD thesis, University of W arwick, 1992

work page 1992

[10] [10]

Nabil M. Mlaiki. Camina triples. Canad. Math. Bull. , 57(1):125–131, 2014

work page 2014

[11] [11]

Isomorphic character tables of nested GV Z -groups

Adriana Nenciu. Isomorphic character tables of nested GV Z -groups. J. Algebra Appl. , 11(2):1250033, 12, 2012

work page 2012

[12] [12]

Nested GVZ-groups

Adriana Nenciu. Nested GVZ-groups. J. Group Theory , 19(4):693–704, 2016

work page 2016

[13] [13]

van der W aall and Eric B

Robert W. van der W aall and Eric B. Kuisch. Homogeneous c haracter induction. II. J. Alge- bra, 170(2):584–595, 1994. Department of Mathematical Sciences, Kent State University , Kent, Ohio 44240, U.S.A. E-mail address : sburket1@kent.edu Department of Mathematical Sciences, Kent State University , Kent, Ohio 44240, U.S.A. E-mail address : lewis@math.kent.edu

work page 1994