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Finite groups of prime-power order obey an explicit formula for their total degree T(G) that confirms a 2008 conjecture.

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load-bearing objection The paper gives an explicit formula for T(G) in prime-power groups that confirms the 2008 conjecture and parallels Hall's class-number result.

arxiv 2607.02048 v1 pith:32PQ22HD submitted 2026-07-02 math.GR

On the total character of a finite group

Thomas Breuer , L\'aszl\'o H\'ethelyi , Burkhard K\"ulshammer , Magdolna Sz\H{o}ke This is my paper
classification math.GR
keywords total charactertotal degreerich subgroupprime power orderclass numberpermutation characterHeffernan-MacHale conjecture
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper defines the total character τ_G of a finite group G as the sum of all its irreducible complex characters and sets T(G) equal to the degree of this character. It introduces the notion of a rich proper subgroup H, meaning that τ_G is contained in the induced permutation character from the trivial character of H, and studies such subgroups when the index is a product of two primes or when G is symmetric or alternating. For groups G whose order is a prime power the authors derive a closed formula for T(G) that is directly analogous to P. Hall's formula for the class number and thereby establishes the truth of the Heffernan-MacHale conjecture. The final section examines groups in which T(G) is small.

Core claim

In the second part we establish a formula for T(G) in the case where the order of G is a prime power. This result is analogous to a formula for the class number of G proved by P. Hall, and it confirms a conjecture by Heffernan and MacHale from 2008.

What carries the argument

The total degree T(G), defined as the value at the identity of the sum of all irreducible complex characters of G.

Load-bearing premise

The standard properties of irreducible characters, their sums, and induced permutation characters extend without modification or additional case distinctions to all groups of prime-power order.

What would settle it

A concrete group G of prime-power order in which the value of T(G) computed directly from its character table differs from the value given by the new formula.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript defines the total character τ_G of a finite group G as the sum of its irreducible complex characters and T(G) := τ_G(1). It studies rich proper subgroups H (those for which τ_G is contained in the induced permutation character (1_H)^G), with emphasis on the case where [G:H] is a product of two primes and on subgroups of symmetric and alternating groups. For groups of prime-power order it derives an explicit formula for T(G) analogous to P. Hall's formula for the class number; this derivation confirms the 2008 conjecture of Heffernan and MacHale. The paper concludes by examining groups in which T(G) is small.

Significance. If the central derivation holds, the explicit formula for T(G) on p-groups constitutes a concrete advance: it supplies a closed-form expression where only a conjecture existed and draws a direct parallel to a classical result of Hall. The confirmation of the Heffernan–MacHale conjecture is therefore a verifiable contribution to the character theory of finite p-groups. The auxiliary study of rich subgroups adds incremental information on the relationship between total characters and permutation characters.

minor comments (3)
  1. [Introduction / §1] The precise meaning of the phrase 'τ_G is contained in (1_H)^G' should be stated explicitly (e.g., via non-negative inner-product coefficients or coefficient-wise inequality) at the first occurrence, rather than left to context.
  2. [Section on prime-power groups] A short table or list of computed values of T(G) for the smallest non-abelian p-groups would make the new formula immediately verifiable and would strengthen the claim that it confirms the 2008 conjecture.
  3. [Section on prime-power groups] The reference to Hall's class-number formula should include a precise citation (paper title, journal, year) at the point where the analogy is first asserted.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of the explicit formula for T(G) in p-groups as a concrete advance confirming the Heffernan–MacHale conjecture, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes a formula for T(G) when |G| is a prime power by applying standard properties of irreducible characters, sums, and induced permutation characters to p-groups. This is presented as analogous to Hall's class-number formula and as confirmation of an external 2008 conjecture by Heffernan and MacHale. No self-citations, fitted parameters renamed as predictions, or self-definitional steps are indicated in the abstract or claims; the derivation rests on uniform character theory that applies without case distinctions specific to the paper's inputs. The central result therefore has independent content relative to its assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard axioms of finite-group representation theory with no free parameters, no invented entities, and no ad-hoc assumptions visible in the abstract.

axioms (2)
  • standard math Standard properties of irreducible complex characters, their sums, and induced permutation characters for finite groups.
    All definitions of τ_G, T(G), and rich subgroups rely on this background theory.
  • standard math Hall's formula for the class number of groups of prime-power order.
    The new formula is presented as analogous to this existing result.

pith-pipeline@v0.9.1-grok · 5726 in / 1362 out tokens · 56811 ms · 2026-07-03T03:07:27.291804+00:00 · methodology

0 comments
read the original abstract

The total character $\tau_G$ of a finite group $G$ is the sum of all irreducible complex characters of $G$, and the total degree of $G$ is $T(G) := \tau_G(1)$. A proper subgroup $H$ of $G$ is rich if $\tau_G$ is ''contained'' in the permutation character $(1_H)^G$. In the first part of this paper, we investigate rich subgroups whose index is a product of two primes. We also consider rich subgroups of symmetric and alternating groups. In the second part we establish a formula for $T(G)$ in the case where the order of $G$ is a prime power. This result is analogous to a formula for the class number of $G$ proved by P. Hall, and it confirms a conjecture by Heffernan and MacHale from 2008. In the last part of the paper, we investigate finite groups $G$ where $T(G)$ is small, in a certain sense.

discussion (0)

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