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arxiv: 2605.04513 · v1 · submitted 2026-05-06 · 🧮 math.RT · math.GR

Zeros of characters and orders of elements in finite groups

Gunter Malle , Gabriel Navarro , Pham Huu Tiep This is my paper

Pith reviewed 2026-05-08 15:44 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords Wilde's conjecturecharacter zeroselement ordersfinite groupsnearly simple groupsirreducible charactersrepresentation theory of finite groups
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The pith

Wilde's conjecture on zeros of characters and orders of elements reduces to a prime-by-prime check on nearly simple groups.

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

The paper aims to establish Wilde's conjecture linking the zeros of irreducible characters to the orders of elements in finite groups. It reduces the full conjecture to a statement about nearly simple groups that can be checked one prime at a time. A sympathetic reader would care because this connection, if true, would show how the character table encodes structural information about the group's elements. The authors verify a strong form of the conjecture in many important cases for nearly simple groups. For primes larger than 5 they confirm the needed statement across most classes of such groups, though some cases await further data on character extensions.

Core claim

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked prime by prime. For these groups, we show that a strong form of Wilde's conjecture holds in many important cases, and for primes p>5 we are able to show the required statement for most classes of nearly simple groups. The few remaining cases, however, seem to require information on extensions of irreducible characters that are not available at the present time.

What carries the argument

The reduction of Wilde's conjecture to a prime-by-prime verifiable statement on nearly simple groups, together with explicit checks for most classes of those groups.

Load-bearing premise

The reduction to nearly simple groups is complete and the explicit verifications performed for the covered classes and primes are accurate.

What would settle it

A single counterexample consisting of a nearly simple group G, a prime p>5, and an element g in G such that some irreducible character of G vanishes at g but p does not divide the order of g, or the converse situation, within one of the classes claimed to be settled.

read the original abstract

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked ``prime by prime". For these groups, we show that a strong form of Wilde's conjecture holds in many important cases, and for primes $p>5$ we are able to show the required statement for most classes of nearly simple groups. The few remaining cases, however, seem to require information on extensions of irreducible characters that are not available at the present time.

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 / 3 minor

Summary. The manuscript investigates Wilde's conjecture on the relationship between zeros of irreducible characters and orders of elements in finite groups. It reduces the conjecture to a prime-by-prime statement on nearly simple groups. The authors establish a strong form of the conjecture in many important cases and, for primes p>5, verify the required statement for most classes of nearly simple groups, while noting that the few remaining cases require currently unavailable information on extensions of irreducible characters.

Significance. The reduction to nearly simple groups, verifiable prime by prime using standard character theory, is a useful structural contribution that decomposes the conjecture without circularity. The explicit verifications in important cases and for p>5 in most nearly simple classes provide concrete progress and identify precise gaps where further data on character extensions would be needed. This case-by-case approach strengthens the field by making the conjecture more amenable to systematic checking.

major comments (1)
  1. [Abstract and reduction section] The central reduction to nearly simple groups is presented as allowing prime-by-prime checks, but the manuscript explicitly states that for the remaining exceptional cases (even when p>5) the required statement cannot be verified due to unavailable data on irreducible character extensions. This limitation is load-bearing for the scope of the claimed progress and should be accompanied by a precise list of the unverified classes in the main text to allow readers to assess the coverage.
minor comments (3)
  1. [Abstract] Clarify in the abstract and introduction what constitutes the 'many important cases' and 'most classes' of nearly simple groups (e.g., by naming the families or citing the relevant theorems) to make the scope of the verifications unambiguous.
  2. [Introduction or §2] Ensure that the exact statement of Wilde's conjecture is recalled verbatim at the start of the reduction argument for reader convenience.
  3. [Conclusion] The paper would benefit from a short concluding section or table summarizing which classes of nearly simple groups are fully verified for each prime range.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the reduction's utility, and the constructive suggestion for improving clarity. We agree that an explicit list of the remaining unverified cases will help readers evaluate the scope of the claimed progress and will incorporate this in the revision.

read point-by-point responses

  1. Referee: [Abstract and reduction section] The central reduction to nearly simple groups is presented as allowing prime-by-prime checks, but the manuscript explicitly states that for the remaining exceptional cases (even when p>5) the required statement cannot be verified due to unavailable data on irreducible character extensions. This limitation is load-bearing for the scope of the claimed progress and should be accompanied by a precise list of the unverified classes in the main text to allow readers to assess the coverage.

    Authors: We agree that the current phrasing in the abstract and reduction section leaves the precise scope of the verifications implicit. In the revised manuscript we will add an explicit list (or table) of the unverified classes of nearly simple groups for p > 5. This list will appear in the reduction section immediately after the statement of the prime-by-prime reduction, enumerating the specific families (e.g., certain extensions of simple groups of Lie type or sporadic groups) where the required character-extension data is currently unavailable. The addition will not change any theorems or proofs but will make the coverage of the results transparent. revision: yes

Circularity Check

0 steps flagged

Reduction to nearly simple groups uses standard character theory without circular dependence on Wilde's conjecture

full rationale

The paper's derivation consists of a reduction of Wilde's conjecture to a prime-by-prime statement on nearly simple groups, followed by direct verification of a strong form in many cases using established tools of finite group character theory. No step equates a derived quantity to its own input by definition, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose justification loops back to the present work. The remaining open cases are explicitly flagged as requiring external data on character extensions that is unavailable, confirming the argument is self-contained where it claims progress and does not manufacture completeness via circular reasoning.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies entirely on established results from the representation theory of finite groups and the classification of nearly simple groups; no new free parameters, axioms beyond standard mathematics, or invented entities are introduced.

axioms (2)
  • standard math Standard axioms and theorems of finite group representation theory, including properties of irreducible characters and their zeros
    The reduction and case analysis presuppose the existing body of character theory for finite groups.
  • domain assumption Known structural properties of nearly simple groups and their character tables
    The prime-by-prime checks rest on the classification and character data of nearly simple groups.

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Reference graph

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