Primitive characters of odd order groups

Claudio Marchi

arxiv: 1906.11207 · v1 · pith:SGKU3MLGnew · submitted 2019-06-26 · 🧮 math.GR · math.RT

Primitive characters of odd order groups

Claudio Marchi This is my paper

Pith reviewed 2026-05-25 14:57 UTC · model grok-4.3

classification 🧮 math.GR math.RT

keywords finite groupsodd orderirreducible charactersprimitive charactersconjugacy classescharacter degreesgroup order

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The pith

In finite groups of odd order, the p-part of any irreducible primitive character degree divides the size of some conjugacy class for each prime p dividing the group order.

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

The paper establishes that for a finite group G of odd order and an irreducible primitive character χ of G, each prime p dividing |G| has a conjugacy class C where the highest power of p dividing χ(1) divides |C|. It further shows that in some classes of such groups the full degree χ(1) divides the size of one conjugacy class. This links the arithmetic of character degrees directly to the sizes of conjugacy classes under the odd-order hypothesis. A reader would care because it supplies a concrete divisibility constraint that must hold for primitive representations in these groups.

Core claim

Let G be a finite group of odd order. We show that if χ is an irreducible primitive character of G then for all primes p dividing the order of G there is a conjugacy class such that the p-part of χ(1) divides the size of that conjugacy class. We also show that for some classes of groups the entire degree of an irreducible primitive character χ divides the size of a conjugacy class.

What carries the argument

The existence, for each prime p dividing |G|, of a conjugacy class whose order is divisible by the p-part of the degree of an irreducible primitive character.

If this is right

The prime-power factors of χ(1) are each controlled by a separate conjugacy class size.
In designated families of odd-order groups the full integer χ(1) divides a single class size.
Any list of possible degrees for primitive characters in odd-order groups must satisfy these divisibility relations.
The result supplies an obstruction to realizing certain integers as primitive character degrees.

Where Pith is reading between the lines

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

The same divisibility might hold for non-primitive irreducible characters under stronger hypotheses on G.
One could test whether the full degree divides a class size in all odd-order groups rather than only selected classes.
The constraint may interact with the fact that odd-order groups are solvable, producing bounds on the possible degrees.

Load-bearing premise

The group must have odd order so that the parity condition guarantees a conjugacy class whose size absorbs the required p-power.

What would settle it

Exhibit one finite group G of odd order, one irreducible primitive character χ of G, and one prime p dividing |G| such that the p-part of χ(1) divides none of the conjugacy class sizes in G.

read the original abstract

Let $G$ be a finite group of odd order. We show that if $\chi$ is an irreducible primitive character of $G$ then for all primes $p$ dividing the order of $G$ there is a conjugacy class such that the $p-$part of $\chi(1)$ divides the size of that conjugacy class. We also show that for some classes of groups the entire degree of an irreducible primitive character $\chi$ divides the size of a conjugacy class.

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

Paper shows p-part of primitive irrep degree divides some class size in odd order groups, but scope is narrow and secondary claim vague.

read the letter

The central new thing is the theorem that if G has odd order and χ is an irreducible primitive character of G, then for every prime p dividing |G| there exists a conjugacy class C with the p-part of χ(1) dividing |C|. The paper also claims that in certain groups the full degree divides some class size. This is a concrete statement that appears not to be a direct restatement of earlier work. The argument benefits from the odd-order assumption, which brings in solvability and lets the authors guarantee the existence of the right class. That part is handled cleanly enough on the surface. The main limitation is the narrow scope. Odd-order groups are all solvable, and primitive characters in solvable groups have a lot of structure already. It is possible that the divisibility follows from standard facts about the kernels or the inducing subgroups rather than requiring a new proof. The secondary claim is too vague as written; some classes of groups needs to be made explicit to be useful. Without seeing the full proof it is hard to judge if there are any gaps in the reasoning, but the statement itself has no obvious internal problems. The paper is aimed at finite group theorists who care about character degrees and conjugacy classes in solvable groups. Someone already working on similar divisibility questions might find the relation interesting, but it is unlikely to change much outside that circle. I would bring it to a reading group only if the group is focused on character theory of solvable groups. I would not cite it in my own work soon because the result is too specialized. Still, the claim is precise enough that a serious editor should send it to referees rather than desk reject.

Referee Report

1 major / 0 minor

Summary. The paper claims that if G is a finite group of odd order and χ is an irreducible primitive character of G, then for every prime p dividing |G| there exists a conjugacy class C such that the p-part of χ(1) divides |C|. It further claims that for certain classes of groups the full degree χ(1) divides the size of some conjugacy class.

Significance. If established, the result would give a new divisibility relation between degrees of primitive irreducible characters and conjugacy class sizes, restricted to the solvable case of odd-order groups. Such relations are uncommon and could be useful in the study of character tables or the structure of degrees in solvable groups. The manuscript supplies no supporting lemmas, derivations, or examples, so the significance cannot be evaluated from the given text.

major comments (1)

[Abstract / entire manuscript] The manuscript consists solely of the abstract statement of the two claims. No lemmas, propositions, proofs, or even a sketch of the argument appear anywhere in the text, rendering it impossible to verify whether the mathematics supports the stated theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract / entire manuscript] The manuscript consists solely of the abstract statement of the two claims. No lemmas, propositions, proofs, or even a sketch of the argument appear anywhere in the text, rendering it impossible to verify whether the mathematics supports the stated theorems.

Authors: The referee correctly observes that the submitted manuscript contains only the abstract and provides no proofs, lemmas, or supporting arguments. This omission prevents verification of the claims. We will revise the submission by including the complete proofs and any necessary supporting material from the full version of the work. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a theorem deriving a conjugacy-class divisibility property for primitive irreducible characters of odd-order groups. The abstract and description present this as a derived result under an explicit parity restriction, with no equations, definitions, or self-citations that reduce the claimed statement to its inputs by construction. The result is conditional on the group order being odd and on the character being primitive and irreducible; these are independent assumptions rather than tautological redefinitions or fitted predictions. No load-bearing self-citation chains or ansatz smuggling are indicated.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard background facts from the character theory of finite groups; the abstract introduces no new parameters, entities, or ad-hoc axioms.

axioms (1)

standard math Standard facts about irreducible characters, induction, and conjugacy classes in finite groups of odd order
The result is stated in the language of these background theorems.

pith-pipeline@v0.9.0 · 5589 in / 1161 out tokens · 41397 ms · 2026-05-25T14:57:14.181730+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Casolo and S

C. Casolo and S. Dolfi , Products of primes in conjugacy class sizes and irreducible ch aracter degrees, Israel Journal of Mathematics, 174 (2009), pp. 403–418

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E. J. Cassell , Conjugacy classes in ﬁnite groups, commuting graphs and cha racter degrees, PhD thesis, University of Birmingham, 2013

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Espuelas , Regular orbits on symplectic modules , Journal of Algebra, 138 (1991), pp

A. Espuelas , Regular orbits on symplectic modules , Journal of Algebra, 138 (1991), pp. 1–12

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Gajendragadkar, A characteristic class of characters of ﬁnite π-separable groups, Journal of algebra, 59 (1979), pp

D. Gajendragadkar, A characteristic class of characters of ﬁnite π-separable groups, Journal of algebra, 59 (1979), pp. 237–259

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Huppert, Character theory of ﬁnite groups , vol

B. Huppert, Character theory of ﬁnite groups , vol. 25, Walter de Gruyter, 1998

work page 1998
[6]

Isaacs, T

I. Isaacs, T. Keller, U. Meierfrankenfeld, and A. Moretó , Fixed point spaces, prim- itive character degrees and conjugacy class sizes , Proceedings of the American Mathematical Society, 134 (2006), pp. 3123–3130

work page 2006
[7]

I. M. Isaacs , Primitive characters, normal subgroups, and M-groups , Mathematische Zeitschrift, 177 (1981), pp. 267–284

work page 1981
[8]

, Characters of solvable groups , American Mathematical Society, 2018

work page 2018
[9]

Karpilovsky , Group Representations, vol

G. Karpilovsky , Group Representations, vol. 4, Elsevier, 2016

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[10]

Loukaki , Hyperbolic modules and cyclic subgroups , Journal of Algebra, 266 (2003), pp

M. Loukaki , Hyperbolic modules and cyclic subgroups , Journal of Algebra, 266 (2003), pp. 34– 50

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[11]

Manz and T

O. Manz and T. R. Wolf , Representations of solvable groups, vol. 185, Cambridge University Press, 1993

work page 1993
[12]

Pacifici, On the number of anisotropic simple submodules in modules wi th a form , Archiv der Mathematik, 84 (2005), pp

E. Pacifici, On the number of anisotropic simple submodules in modules wi th a form , Archiv der Mathematik, 84 (2005), pp. 1–10

work page 2005
[13]

Turull , Supersolvable automorphism groups of solvable groups , Mathematische Zeitschrift, 183 (1983), pp

A. Turull , Supersolvable automorphism groups of solvable groups , Mathematische Zeitschrift, 183 (1983), pp. 47–73

work page 1983
[14]

T. R. Wolf , Solvable and nilpotent subgroups of GL(n, qm), Canadian Journal of Mathematics, 34 (1982), pp. 1097–1111

work page 1982
[15]

, Sylow p-subgroups of p-solvable subgroups of GL(n, p) , Archiv der Mathematik, 43 (1984), pp. 1–10

work page 1984
[16]

Yang, Regular orbits of ﬁnite primitive solvable groups, ii , Journal of Algebra, 341 (2011), pp

Y. Yang, Regular orbits of ﬁnite primitive solvable groups, ii , Journal of Algebra, 341 (2011), pp. 23–34. School of Mathematics, University of Manchester, Oxford Road, M13 9PL, Manch- ester, United Kingdom E-mail address : claudio.marchi@manchester.ac.uk

work page 2011

[1] [1]

Casolo and S

C. Casolo and S. Dolfi , Products of primes in conjugacy class sizes and irreducible ch aracter degrees, Israel Journal of Mathematics, 174 (2009), pp. 403–418

work page 2009

[2] [2]

E. J. Cassell , Conjugacy classes in ﬁnite groups, commuting graphs and cha racter degrees, PhD thesis, University of Birmingham, 2013

work page 2013

[3] [3]

Espuelas , Regular orbits on symplectic modules , Journal of Algebra, 138 (1991), pp

A. Espuelas , Regular orbits on symplectic modules , Journal of Algebra, 138 (1991), pp. 1–12

work page 1991

[4] [4]

Gajendragadkar, A characteristic class of characters of ﬁnite π-separable groups, Journal of algebra, 59 (1979), pp

D. Gajendragadkar, A characteristic class of characters of ﬁnite π-separable groups, Journal of algebra, 59 (1979), pp. 237–259

work page 1979

[5] [5]

Huppert, Character theory of ﬁnite groups , vol

B. Huppert, Character theory of ﬁnite groups , vol. 25, Walter de Gruyter, 1998

work page 1998

[6] [6]

Isaacs, T

I. Isaacs, T. Keller, U. Meierfrankenfeld, and A. Moretó , Fixed point spaces, prim- itive character degrees and conjugacy class sizes , Proceedings of the American Mathematical Society, 134 (2006), pp. 3123–3130

work page 2006

[7] [7]

I. M. Isaacs , Primitive characters, normal subgroups, and M-groups , Mathematische Zeitschrift, 177 (1981), pp. 267–284

work page 1981

[8] [8]

, Characters of solvable groups , American Mathematical Society, 2018

work page 2018

[9] [9]

Karpilovsky , Group Representations, vol

G. Karpilovsky , Group Representations, vol. 4, Elsevier, 2016

work page 2016

[10] [10]

Loukaki , Hyperbolic modules and cyclic subgroups , Journal of Algebra, 266 (2003), pp

M. Loukaki , Hyperbolic modules and cyclic subgroups , Journal of Algebra, 266 (2003), pp. 34– 50

work page 2003

[11] [11]

Manz and T

O. Manz and T. R. Wolf , Representations of solvable groups, vol. 185, Cambridge University Press, 1993

work page 1993

[12] [12]

Pacifici, On the number of anisotropic simple submodules in modules wi th a form , Archiv der Mathematik, 84 (2005), pp

E. Pacifici, On the number of anisotropic simple submodules in modules wi th a form , Archiv der Mathematik, 84 (2005), pp. 1–10

work page 2005

[13] [13]

Turull , Supersolvable automorphism groups of solvable groups , Mathematische Zeitschrift, 183 (1983), pp

A. Turull , Supersolvable automorphism groups of solvable groups , Mathematische Zeitschrift, 183 (1983), pp. 47–73

work page 1983

[14] [14]

T. R. Wolf , Solvable and nilpotent subgroups of GL(n, qm), Canadian Journal of Mathematics, 34 (1982), pp. 1097–1111

work page 1982

[15] [15]

, Sylow p-subgroups of p-solvable subgroups of GL(n, p) , Archiv der Mathematik, 43 (1984), pp. 1–10

work page 1984

[16] [16]

Yang, Regular orbits of ﬁnite primitive solvable groups, ii , Journal of Algebra, 341 (2011), pp

Y. Yang, Regular orbits of ﬁnite primitive solvable groups, ii , Journal of Algebra, 341 (2011), pp. 23–34. School of Mathematics, University of Manchester, Oxford Road, M13 9PL, Manch- ester, United Kingdom E-mail address : claudio.marchi@manchester.ac.uk

work page 2011