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arxiv: 1907.02898 · v1 · pith:KGTVYT5Fnew · submitted 2019-07-05 · 🧮 math.GR · math.CO

On the Intersection Numbers of Finite Groups

Kassie Archer , Humberto Bautista Serrano , Kayla Cook , L.-K. Lauderdale , Yansy Perez , Vincent Villalobos This is my paper

Pith reviewed 2026-05-25 01:47 UTC · model grok-4.3

classification 🧮 math.GR math.CO
keywords intersection numberFrattini subgroupmaximal subgroupsnilpotent groupsfinite groupscovering number
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The pith

Nontrivial finite nilpotent groups admit an exact formula for the minimum number of maximal subgroups whose intersection is the Frattini subgroup.

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

The paper defines the intersection number ι(G) of a nontrivial finite group G as the smallest collection of maximal subgroups whose common intersection equals the Frattini subgroup Φ(G). It supplies a closed-form expression for this number in every case where G is nilpotent. The same quantity is computed explicitly for several infinite families of non-nilpotent groups. The definition supplies a dual counterpart to the covering number σ(G), which instead measures the smallest number of proper subgroups needed to cover the whole group.

Core claim

For a nontrivial finite group G, ι(G) is the minimum number of maximal subgroups whose intersection equals the Frattini subgroup Φ(G). When G is nilpotent this number is given by an exact formula; the paper also determines the value for selected infinite families of non-nilpotent groups and introduces a natural generalization of the invariant.

What carries the argument

The intersection number ι(G), defined as the size of the smallest set of maximal subgroups whose intersection equals Φ(G).

If this is right

  • ι(G) is now known exactly for every nontrivial finite nilpotent group.
  • The intersection numbers of certain infinite non-nilpotent families are completely determined.
  • A natural generalization of ι(G) is available for further study.
  • Several open questions about these invariants are posed for future work.

Where Pith is reading between the lines

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

  • The formula for nilpotent groups may extend to solvable groups whose maximal subgroups are sufficiently well understood.
  • ι(G) could be compared directly with the minimal number of generators of the Frattini quotient G/Φ(G).
  • Computing ι(G) for symmetric or alternating groups might reveal patterns tied to their subgroup lattices.

Load-bearing premise

The Frattini subgroup equals the intersection of every maximal subgroup, so that a minimal number of them can achieve exactly that intersection.

What would settle it

A single nontrivial finite nilpotent group G together with an explicit list of its maximal subgroups whose intersections fail to match the claimed formula for ι(G).

read the original abstract

The covering number of a nontrivial finite group $G$, denoted $\sigma(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. For a nontrivial finite group $G$, we define the intersection number of $G$, denoted $\iota(G)$, to be the minimum number of maximal subgroups whose intersection equals the Frattini subgroup of $G$. We elucidate some basic properties of this invariant, and give an exact formula for $\iota(G)$ when $G$ is a nontrivial finite nilpotent group. In addition, we determine the intersection numbers of a few infinite families of non-nilpotent groups. We conclude by discussing a generalization of the intersection number of a nontrivial finite group and pose some open questions about these invariants.

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 / 2 minor

Summary. The paper defines the intersection number ι(G) for a nontrivial finite group G as the minimum number of maximal subgroups whose intersection equals the Frattini subgroup Φ(G). It states an exact formula for ι(G) when G is nilpotent (derived from the primary decomposition into Sylow subgroups), computes ι(G) explicitly for several infinite families of non-nilpotent groups, discusses a generalization of the invariant, and poses open questions.

Significance. If the stated formula holds, the work introduces a well-defined dual invariant to the covering number σ(G) and supplies an exact, parameter-free expression for all nontrivial finite nilpotent groups via their Sylow decomposition; this is a clear strength. The explicit computations for non-nilpotent families provide concrete data points that can be checked directly against the definition.

minor comments (2)
  1. [Abstract] Abstract: the claim that an exact formula is given for nilpotent groups is not accompanied by even a brief statement of the formula itself; adding one sentence summarizing the formula would make the abstract self-contained.
  2. The generalization section and open questions would benefit from explicit cross-references to the earlier definitions of ι(G) and the nilpotent formula to clarify how the generalization extends the main results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. The referee's description of the paper is accurate. No specific major comments are listed in the report, so we have no point-by-point responses to provide. We will incorporate any minor editorial corrections in the revised version.

Circularity Check

0 steps flagged

No significant circularity; definition and formulas are independent

full rationale

The paper introduces ι(G) as a freshly defined minimum (the smallest number of maximal subgroups intersecting exactly to the standard Frattini subgroup Φ(G)). The exact formula for nilpotent groups is derived from the primary decomposition G = P1 × ⋯ × Pr together with the correspondence of maximal subgroups to those in the Sylow factors; this is a direct group-theoretic computation, not a reduction to fitted parameters or self-citations. Results for the listed non-nilpotent families are explicit computations. No load-bearing step matches any of the enumerated circularity patterns; the central claims rest on external standard theorems about Φ(G) and Sylow structure rather than on the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard fact that Φ(G) is the intersection of all maximal subgroups; no free parameters are fitted and no new entities are postulated.

axioms (1)
  • domain assumption The Frattini subgroup Φ(G) equals the intersection of all maximal subgroups of any finite group G.
    This classical theorem is required for the definition of ι(G) to be meaningful as the minimal number of maximal subgroups achieving exactly that intersection.

pith-pipeline@v0.9.0 · 5691 in / 1216 out tokens · 21656 ms · 2026-05-25T01:47:19.926700+00:00 · methodology

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Reference graph

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