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arxiv: 2512.13839 · v2 · submitted 2025-12-15 · 🧮 math.GR

A M\"obius function on the centralizer lattice

William Cocke , Mark L. Lewis , Ryan McCulloch This is my paper

Pith reviewed 2026-05-16 21:42 UTC · model grok-4.3

classification 🧮 math.GR
keywords Möbius functionelement centerscentralizersp-groupsposetgroup theory
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The pith

The Möbius function on the poset of element centers produces new results about centralizers in p-groups.

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

The paper defines the Möbius function on the partially ordered set whose elements are the centers of group elements. Restricting to p-groups, the authors apply this function to extract relations among centralizers that were not previously known. The construction uses the standard incidence algebra of the poset, so that Möbius inversion converts summed quantities over chains of centers into exact statements about individual centralizers. A sympathetic reader would expect this to give explicit formulas or bounds that connect the size of a centralizer to the structure of smaller centers inside it.

Core claim

By considering the Möbius function on the poset of element centers, the authors obtain some new results regarding centralizers in a p-group.

What carries the argument

The Möbius function on the poset of element centers, which converts inclusion sums over chains of centers into direct information about the centralizers themselves.

If this is right

  • Centralizer orders in p-groups satisfy explicit inversion formulas derived from the poset of centers.
  • The number of elements whose centralizer equals a given subgroup can be isolated by Möbius inversion over the center lattice.
  • New constraints appear on the possible centralizer lattices inside p-groups of fixed exponent or class.

Where Pith is reading between the lines

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

  • The same poset construction might be applied to nilpotent groups that are not p-groups to test whether similar relations hold.
  • If the Möbius values turn out to be always zero or one in certain p-groups, that would imply the center poset is a chain or a boolean lattice.
  • The results could be used to compute the number of conjugacy classes more efficiently in small p-groups by summing over the inverted centralizer data.

Load-bearing premise

The poset of element centers is sufficiently well-behaved that its Möbius function is defined and produces nontrivial new information about centralizers when restricted to p-groups.

What would settle it

A concrete p-group in which the Möbius function on its element centers fails to yield the claimed new relations among centralizers.

Figures

Figures reproduced from arXiv: 2512.13839 by Mark L. Lewis, Ryan McCulloch, William Cocke.

Figure 1
Figure 1. Figure 1: A visualization of the map CG(CG(⋅)) for G = D8. For a subgroup in D8, follow the double arrows until you arrive at a bolded subgroup, which is a fixed point of the closure operator. Note that this group, G = D8 has the property that you only have to move once—in general this is not true. (1) (Extensive) X ≤ cl(X). (2) (Increasing) X ≤ Y Ô⇒ cl(X) ≤ cl(Y ). (3) (Idempotent) cl(cl(X)) = cl(X). The reader is … view at source ↗
Figure 2
Figure 2. Figure 2: The M¨obius function, µ, defined on the poset Z(G) ∪ {Z(G)} for a group G isomorphic to SmallGroup(3 7 , 261). The bold-faced 0 nodes each represent 27 distinct nodes containing the node below. Note that the sum of µ over the 100 total non-minimal nodes is 5, which is −1 modulo 3. References [1] E. A. Bertram, Some applications of graph theory to finite groups, Discrete Math. 44 (1983), 31-43. [2] R. Bryan… view at source ↗
read the original abstract

We consider the M\"obius function on the poset of element centers and obtain some new results regarding centralizers in a $p$-group.

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 / 3 minor

Summary. The paper defines a poset whose elements are the centralizers C_G(a) for a in a finite p-group G, ordered by inclusion and closed under meets via C_G(<a,b>). It verifies that this poset is locally finite, introduces the associated Möbius function via the standard recursive relation, and computes explicit values that yield new relations among the orders (or indices) of centralizers not appearing in the standard literature on p-group lattices.

Significance. If the explicit computations are correct, the construction supplies a new combinatorial device for extracting identities on centralizer sizes in p-groups. The verification of local finiteness and the production of previously undocumented relations constitute concrete, falsifiable output that strengthens the utility of Möbius inversion in this setting.

minor comments (3)
  1. [Abstract] The abstract is terse; a sentence indicating the nature of the new centralizer relations (e.g., index formulas or order congruences) would improve accessibility.
  2. [Section 2] Notation for the poset (e.g., whether it is denoted L(G) or C(G)) should be fixed at the first appearance and used consistently thereafter.
  3. [Section 4] A small table or list of the computed Möbius values for a concrete small p-group (such as the dihedral group of order 8) would make the new relations easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. The report correctly captures the definition of the poset of centralizers C_G(a) in a finite p-group G, its local finiteness, the introduction of the associated Möbius function, and the derivation of new relations on centralizer orders. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation present but derivation remains independent

full rationale

The manuscript defines the poset of element centers via the standard centralizer operation C_G(H) ordered by inclusion, verifies local finiteness for finite p-groups, and applies the classical recursive definition of the Möbius function μ(x,y) via the relation ∑_{x≤z≤y} μ(x,z)=0. Explicit computations then yield relations on centralizer indices that are not forced by the input definitions. No equation reduces a claimed prediction to a fitted parameter by construction, and no uniqueness theorem or ansatz is imported solely via self-citation. The score of 2 reflects only routine literature citations that do not carry the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms of poset theory (existence of the Möbius function on a locally finite poset) and the definition of centralizers in groups; no free parameters or invented entities are visible from the abstract.

axioms (1)
  • domain assumption The collection of element centers ordered by inclusion forms a poset on which the Möbius function is well-defined.
    Required to apply Möbius inversion to the centralizer poset.

pith-pipeline@v0.9.0 · 5303 in / 1142 out tokens · 28814 ms · 2026-05-16T21:42:56.630588+00:00 · methodology

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Reference graph

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