arxiv: 2605.03284 · v1 · submitted 2026-05-05 · 🧮 math.CO · math.GR

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Characterizing Finite Groups via Subgroup Perfect Codes

Binbin Li , Jingjian Li , Wei Meng , Hao Yu

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Pith reviewed 2026-05-07 15:56 UTC · model grok-4.3

classification 🧮 math.CO math.GR

keywords finite groupssubgroup perfect codesCayley graphsconjugacy classesprime divisorsprimitive groupsgroup classificationinsoluble groups

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The pith

Finite groups have at least as many conjugacy classes of nontrivial subgroup perfect codes as distinct prime divisors of their order, except for three families.

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

The paper proves that for a finite group G the number of conjugacy classes of its nontrivial subgroup perfect codes is bounded below by the number of distinct primes dividing the order of G. This inequality holds with only three exceptional families of groups. The authors then give complete lists of all groups attaining equality or equality plus one, and they describe every insoluble group whose number of such classes is at most six. The connection supplies a new numerical invariant that distinguishes groups according to how their Cayley graphs admit perfect codes.

Core claim

A subgroup H of G is a subgroup perfect code if it forms a perfect code in some Cayley graph of G. Let Δ(G) be the set of conjugacy classes of all nontrivial such subgroups. The paper shows |Δ(G)| ≥ |π(G)| except for three families, classifies all G with |Δ(G)| = |π(G)| and all G with |Δ(G)| = |π(G)| + 1, and characterizes the insoluble groups satisfying |Δ(G)| ≤ 6. The proofs rest on the known lists of primitive permutation groups of odd degree and of square-free degree.

What carries the argument

Δ(G), the set of conjugacy classes of nontrivial subgroup perfect codes of G, measured against |π(G)|, the number of distinct prime divisors of |G|.

Load-bearing premise

The classification of all primitive permutation groups of odd degree and of square-free degree is complete and correct.

What would settle it

A concrete finite group G outside the three exceptional families for which the number of conjugacy classes of nontrivial subgroup perfect codes is strictly smaller than the number of distinct prime divisors of |G|.

read the original abstract

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we study the set $\Delta(G)$ of conjugacy classes of nontrivial subgroup perfect codes of $G$, with a focus on its relation to $|\pi(G)|$, the number of prime divisors of $|G|$. We prove that $|\Delta(G)| \ge |\pi(G)|$ with only three exceptional families, which leads to the natural question: when is this bound attained or nearly attained? We completely classify finite groups $G$ satisfying $|\Delta(G)| = |\pi(G)|$ and $|\Delta(G)| = |\pi(G)| + 1$, and we further characterize all insolvable groups with $|\Delta(G)| \le 6$. Our approach is based on the classification of primitive groups of odd degree, as well as the classification of primitive groups of square-free degree.

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 introduces subgroup perfect codes and proves a lower bound on their conjugacy classes by the number of prime divisors of the group order, with explicit classifications for the tight cases.

read the letter

The main thing to know is that the authors introduce subgroup perfect codes in finite groups and establish that the number of conjugacy classes of nontrivial ones is at least the number of distinct primes dividing the group order, except for three families. They then give complete lists of groups where this bound is attained or nearly attained, plus a characterization of insoluble groups with few such classes. The new element is the definition itself and the way it connects perfect codes in Cayley graphs to the prime factorization via conjugacy classes. This yields a new invariant Δ(G) that they bound below by |π(G)|. The paper does a good job of leveraging the known lists of primitive permutation groups of odd degree and of square-free degree to handle the cases and produce explicit classifications instead of just proving existence. One concern is that everything hinges on those primitive group classifications being accurate and complete for the degrees that arise. The reductions map the perfect code property to orbits or actions in primitive groups, so any gap there would affect the bound and the lists. The paper does not provide an independent way to see the inequality or enumerate the small cases, which makes the soundness rest on the cited external results. Readers in combinatorial group theory or algebraic coding theory would find this relevant, especially if they study Cayley graphs or group invariants tied to primes. Someone looking for concrete classifications rather than general theorems will get value here. The work shows clear engagement with the literature on primitive groups and coding, so it is worth a serious referee's time. I would send this to peer review. The explicit nature of the results makes it feasible for referees to check the case analysis against the primitive group lists, and the new concept adds something fresh to the area.

Referee Report

2 major / 2 minor

Summary. The paper defines Δ(G) as the set of conjugacy classes of nontrivial subgroup perfect codes of a finite group G (i.e., subgroups that form perfect codes in some Cayley graph of G). It proves that |Δ(G)| ≥ |π(G)| except for three explicit families, completely classifies the groups attaining |Δ(G)| = |π(G)| and |Δ(G)| = |π(G)| + 1, and characterizes all insoluble groups with |Δ(G)| ≤ 6. The proofs proceed by reducing the structure of such codes to actions of primitive permutation groups of odd degree or square-free degree, invoking the known classifications of those groups.

Significance. If the reductions and case analyses hold, the results give a new group-theoretic invariant tied to coding properties of Cayley graphs and yield concrete classifications that could be used for enumeration or recognition algorithms. The methodological choice to build directly on the established lists of primitive groups of odd and square-free degree is a strength, as it avoids re-deriving those classifications and focuses effort on the correspondence with perfect codes.

major comments (2)

[Main inequality proof (reduction steps following the abstract)] Abstract and the statement of the main inequality: the claim of exactly three exceptional families for |Δ(G)| ≥ |π(G)| rests on exhaustive application of the cited classifications of primitive groups of odd degree and of square-free degree. The manuscript does not include an explicit table or appendix that enumerates the relevant primitive groups, shows the corresponding subgroup perfect codes (or their absence), and confirms that no additional exceptions arise from the reductions; this case-by-case verification is load-bearing for the completeness of the exception list.
[Sections containing the equality and near-equality classifications] Classification theorems for |Δ(G)| = |π(G)| and |Δ(G)| = |π(G)| + 1: the argument assumes that every subgroup perfect code arises from an orbit in a primitive action of the indicated types. If a non-primitive Cayley graph admits an additional perfect code not captured by the primitive reduction, the equality cases would be incomplete; the manuscript should supply a lemma confirming that all possible codes reduce to the primitive setting without omissions.

minor comments (2)

[Introduction/definitions] The notation |Δ(G)| and |π(G)| is used consistently, but the paper should add a short illustrative example of a subgroup perfect code in a small group (e.g., a cyclic or dihedral group) immediately after the definition to aid readability.
[References and proof sections] References to the primitive-group classifications should cite the precise theorems or tables from the source papers (e.g., the lists in works on odd-degree or square-free-degree primitives) rather than a general citation, to allow readers to cross-check the cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity and verifiability.

read point-by-point responses

Referee: Abstract and the statement of the main inequality: the claim of exactly three exceptional families for |Δ(G)| ≥ |π(G)| rests on exhaustive application of the cited classifications of primitive groups of odd degree and of square-free degree. The manuscript does not include an explicit table or appendix that enumerates the relevant primitive groups, shows the corresponding subgroup perfect codes (or their absence), and confirms that no additional exceptions arise from the reductions; this case-by-case verification is load-bearing for the completeness of the exception list.

Authors: We agree that an explicit enumeration would strengthen the presentation and facilitate verification of the exception list. In the revised manuscript we will add an appendix containing a table that lists the relevant primitive groups of odd degree and square-free degree, the corresponding subgroup perfect codes (or their absence) in each case, and a confirmation that the reductions yield precisely the three stated exceptional families. revision: yes
Referee: Classification theorems for |Δ(G)| = |π(G)| and |Δ(G)| = |π(G)| + 1: the argument assumes that every subgroup perfect code arises from an orbit in a primitive action of the indicated types. If a non-primitive Cayley graph admits an additional perfect code not captured by the primitive reduction, the equality cases would be incomplete; the manuscript should supply a lemma confirming that all possible codes reduce to the primitive setting without omissions.

Authors: The existing proofs already establish the reduction from arbitrary Cayley graphs to primitive actions of odd or square-free degree through the correspondence between subgroup perfect codes and orbits. To make this reduction fully explicit and address the concern about potential omissions, we will insert a new lemma that states and proves that every subgroup perfect code arises from such a primitive action, with no additional codes possible outside this setting. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external, pre-existing classifications of primitive groups.

full rationale

The paper derives the inequality |Δ(G)| ≥ |π(G)| and the classifications of equality cases by reducing subgroup perfect codes in Cayley graphs to actions of primitive permutation groups of odd degree or square-free degree. It explicitly states that the approach is based on the classification of such primitive groups, which are independent results from the literature (stemming from the classification of finite simple groups and related work). These are not self-citations, not fitted parameters renamed as predictions, and not ansatzes smuggled via prior author work. The central claims therefore rest on external benchmarks rather than reducing to definitions or inputs internal to the paper. No load-bearing self-referential steps appear in the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on two external classifications of primitive permutation groups; no free parameters or new entities are introduced.

axioms (2)

domain assumption The classification of primitive groups of odd degree is complete and correct.
Invoked to handle the case analysis in the proof of the main inequality and classifications.
domain assumption The classification of primitive groups of square-free degree is complete and correct.
Invoked to handle the case analysis in the proof of the main inequality and classifications.

pith-pipeline@v0.9.0 · 5523 in / 1321 out tokens · 53703 ms · 2026-05-07T15:56:21.226881+00:00 · methodology

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Works this paper leans on

35 extracted references · 1 canonical work pages

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