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arxiv: 2604.10100 · v1 · submitted 2026-04-11 · 🧮 math.RT · math.GR

Non-solvable groups whose non-linear character degrees have the same number of different prime divisors

Junying Guo , Yanjun Liu , Ziyi Wu , Di Xiao This is my paper

Pith reviewed 2026-05-10 16:03 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords character degreesnon-solvable groupsprime divisorsfinite groupsHuppert conjecturelinear groupsalternating groups
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The pith

Finite non-solvable groups whose non-linear character degrees share the same number of distinct prime divisors are, up to an abelian direct factor, one of six explicit families.

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

The paper classifies all finite non-solvable groups in which every non-linear irreducible character degree is divisible by exactly the same number of distinct primes. Building from a known result that solvable groups with this property must be meta-abelian, the authors prove that the non-solvable examples reduce to a short explicit list. If the classification holds, then the primes dividing any irreducible character degree of such a group can only be 2, 3, 5, or 7, and Huppert's ρ-σ conjecture is verified for the entire class.

Core claim

Up to an abelian direct factor, the finite non-solvable groups whose non-linear character degrees have the same number of different prime divisors are exactly L_2(4), L_2(8), A_7, S_7, the central product of a cyclic 3-group with 3.A_7, or the semi-direct product of A_7 by a cyclic 2-group that acts non-trivially on A_7 by conjugation.

What carries the argument

The uniform prime-divisor-count condition on the set of non-linear irreducible character degrees, which restricts the possible groups to the listed families.

If this is right

  • The only primes that divide irreducible character degrees of such groups are 2, 3, 5, and 7.
  • Huppert's ρ-σ conjecture holds for all groups satisfying the degree condition.
  • The possible groups are limited to specific linear, alternating, and almost-simple extensions of A_7.

Where Pith is reading between the lines

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

  • The classification supplies a complete list of base cases that could be used to test related conjectures on character degrees for larger or infinite families.
  • Similar restrictions on the prime factors of character degrees may isolate other small collections of almost-simple groups.
  • The result separates the non-solvable case cleanly from the solvable meta-abelian case, suggesting the uniform-count condition behaves differently across solvability.

Load-bearing premise

No other finite non-solvable groups outside the enumerated families satisfy the condition that all non-linear character degrees have the same number of distinct prime divisors.

What would settle it

A non-solvable group not isomorphic to any group in the listed families (after removing an abelian direct factor) in which every non-linear irreducible character degree is divisible by the same number of distinct primes.

read the original abstract

By a result of Noritzsch, a finite solvable group whose non-linear character degrees have the same set of prime divisors is meta-abelian. In this note we investigate finite non-solvable groups whose non-linear character degrees have the same number of different prime divisors, and show that up to an abelian direct factor, such groups are exactly $L_2(4), L_2(8), A_7, S_7$, the central product of a cyclic $3$-group with $3.A_7$, or the semi-direct product of $A_7$ by a cyclic $2$-group $\langle a\rangle$ such that $a$ non-trivially acts on $A_7$ by conjugation. As consequence, we show that only the primes $2,3,5,7$ may occur as prime divisors of their irreducible character degrees, and that Huppert's $\rho$-$\sigma$ conjecture holds for them.

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

0 major / 4 minor

Summary. The manuscript investigates finite non-solvable groups G where all non-linear irreducible character degrees have the same number of distinct prime divisors. It classifies such groups up to an abelian direct factor as L_2(4), L_2(8), A_7, S_7, the central product of a cyclic 3-group with 3.A_7, or the semidirect product A_7 ⋊ ⟨a⟩ (⟨a⟩ cyclic of order 2 acting non-trivially on A_7). Consequences include that only primes 2, 3, 5, 7 divide the character degrees and that Huppert's ρ-σ conjecture holds for these groups. The argument uses a minimal counterexample G, restricts possible composition factors using the constant-ω condition, and verifies the property on the remaining candidates via their character tables.

Significance. This provides a non-solvable counterpart to Noritzsch's meta-abelian result for solvable groups under a similar (but weaker) condition. The classification is explicit and finite, allowing the consequences to follow immediately. The case analysis on non-abelian composition factors and direct checks on small groups like A7 and L2(8) are standard techniques in the field and appear to cover the possibilities effectively.

minor comments (4)
  1. The contrast with Noritzsch's theorem could be made clearer by stating explicitly that the solvable case requires the same set of primes rather than just the same cardinality.
  2. The description of the semidirect product could specify the homomorphism from the cyclic 2-group to Out(A7) to distinguish it from S7 if necessary.
  3. A short recall of the statement of Huppert's ρ-σ conjecture would help readers understand the second consequence without looking it up.
  4. The reduction step that forces the non-linear degrees to be supported only on primes {2,3,5,7} is central; the manuscript should clarify in which section this is proved and whether it uses the classification of simple groups with character degrees having at most two prime factors or a similar result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment, including the recommendation for minor revision. The provided summary accurately captures the main results and methods.

Circularity Check

0 steps flagged

No significant circularity; standard classification via minimal counterexample

full rationale

The paper cites Noritzsch's external theorem for solvable groups, then assumes a minimal non-solvable counterexample and performs case analysis on its non-abelian composition factors. The constant-ω condition on non-linear degrees restricts candidates to a short explicit list (L2(4), L2(8), A7 and listed extensions), which are verified directly from character tables. No step reduces by definition to its own input, no parameters are fitted then renamed as predictions, and no load-bearing premise rests on self-citation chains. The argument is self-contained against external benchmarks such as the classification of finite simple groups.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification rests on the standard axioms of finite group theory and the representation theory of finite groups, together with the cited Noritzsch theorem; no free parameters, no new postulated entities.

axioms (1)
  • standard math Standard axioms and theorems of finite group theory and character theory, including Noritzsch's result on solvable groups
    Invoked to reduce the non-solvable case to known simple groups and their extensions.

pith-pipeline@v0.9.0 · 5476 in / 1353 out tokens · 55918 ms · 2026-05-10T16:03:51.876860+00:00 · methodology

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Reference graph

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