Covering by Centralizers

Mark L. Lewis; Ryan McCulloch

arxiv: 2512.02211 · v2 · submitted 2025-12-01 · 🧮 math.GR

Covering by Centralizers

Mark L. Lewis , Ryan McCulloch This is my paper

Pith reviewed 2026-05-17 02:03 UTC · model grok-4.3

classification 🧮 math.GR

keywords centralizerscoversfinite groupsp-groupsmaximal subgroupsminimal subgroups

0 comments

The pith

Maximal centralizers and minimal centralizers each cover any finite group.

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

The paper shows that in a finite group the centralizers maximal under inclusion by set containment form a cover, so their union equals the whole group. It proves the same for the minimal centralizers under the same ordering. For the special case of nonabelian p-group F-groups the distinct nontrivial centralizers number one more than a multiple of p. These statements give a uniform way to reach every group element via centralizers selected by extremal size.

Core claim

The collection of all maximal centralizers covers the group and the collection of all minimal centralizers also covers the group. In nonabelian p-group F-groups the number of distinct nontrivial centralizers is congruent to 1 modulo p.

What carries the argument

The partial order of centralizers by inclusion, whose maximal and minimal members are extracted to form the covers.

If this is right

Every element of a finite group lies inside at least one maximal centralizer.
Every element of a finite group lies inside at least one minimal centralizer.
In nonabelian F-group p-groups the distinct nontrivial centralizers are counted by a formula congruent to 1 mod p.

Where Pith is reading between the lines

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

The two covers may interact with the center or with conjugacy-class sizes in ways that refine the counting argument.
Similar extremal-centralizer covers might be examined in solvable or nilpotent groups without the F-group hypothesis.

Load-bearing premise

The groups are finite, and the congruence statement further requires them to be F-groups that are nonabelian p-groups.

What would settle it

A single finite group containing an element outside every maximal centralizer, or a nonabelian F-group p-group whose count of distinct nontrivial centralizers fails to be congruent to 1 modulo p.

Figures

Figures reproduced from arXiv: 2512.02211 by Mark L. Lewis, Ryan McCulloch.

**Figure 1.** Figure 1: A Hasse diagram showing the centralizers and centers of S4. We close the paper with two interesting observations about F-groups that are p-groups. Theorem 5.7. Let G be p-group that is an F-group and is not a CA-group. There exists a nonabelian centralizer C with exp(C) = exp(Z(C)). Proof. Among all nonabelian proper centralizers in G, choose one whose center has the largest exponent, call this centralizer… view at source ↗

read the original abstract

In this paper, we consider covers of finite groups by centralizers of elements. We show that the set of centralizers that are maximal under the partial ordering form a cover of the group. We also show that the set of centralizers that are minimal under the partial ordering form a cover of the group. We show for $F$-groups that are nonabelian $p$-groups that the number of distinct nontrivial centralizers is congruent to $1$ modulo $p$.

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

Maximal centralizer covers are immediate from finiteness while the minimal covers and F-group congruence are the more substantive claims.

read the letter

The punchline on this one is that the maximal centralizers cover any finite group by a quick poset argument, the minimal centralizers do too, and there's a congruence result for the number of nontrivial centralizers in certain p-groups. The paper takes the inclusion poset of centralizers in a finite group G and shows the maximal elements form a cover, meaning every element of G is in some maximal centralizer. This holds because the poset is finite and nonempty, so chains have maximal elements. For minimal centralizers, they argue that for each g you can find a minimal centralizer containing it. The F-group part restricts to nonabelian p-groups and gets the count congruent to 1 mod p, which is probably from some orbit or fixed-point counting. It does well by keeping the claims short and direct. The abstract lays out exactly what is proved without extra fluff. The soft spots are that the maximal cover follows almost right away from basic facts about finite posets, so it might not add much new insight there. The minimal cover and the congruence are more interesting, but I would want to see the definition of F-group and how the proof goes to judge the depth. If the proofs are short and elementary, the whole thing is a modest note rather than a big result. This paper is for group theorists working on finite groups, centralizers, or subgroup covers. Someone already familiar with p-group enumeration or centralizer lattices could get value from the specific count. I would recommend sending it for peer review. The claims look consistent and worth checking the details on, even if the maximal part is straightforward.

Referee Report

1 major / 2 minor

Summary. The paper considers the poset of centralizers of elements in a finite group G, ordered by inclusion. It claims to prove that the maximal centralizers form a cover of G (their union is G) and that the minimal centralizers likewise form a cover of G. It further claims that when G is a nonabelian p-group that is an F-group, the number of distinct nontrivial centralizers is congruent to 1 modulo p.

Significance. If the claims hold, the work records elementary but potentially useful facts about the centralizer poset in finite groups: every element lies in a maximal centralizer (immediate from finiteness) and in a minimal centralizer (requiring a short argument that a minimal element exists inside the subposet of centralizers containing a fixed element). The congruence supplies a modular counting constraint for centralizers in a restricted class of p-groups. The arguments appear to rest on standard definitions and the finiteness of the poset rather than any self-referential or parameter-tuned construction.

major comments (1)

[Proof that minimal centralizers cover the group] The argument that minimal centralizers cover G is less immediate than the maximal case. The manuscript should explicitly verify, for an arbitrary g, that the subposet of centralizers containing C(g) has a minimal element that is still a centralizer of some group element (rather than merely a minimal member of a larger collection).

minor comments (2)

[Section introducing F-groups and the congruence] The precise definition of an F-group must appear in the body (not only the abstract) before the congruence statement is stated, since the result is restricted to this subclass.
[Introduction or first theorem] The manuscript should note explicitly that the maximal-centralizer cover follows at once from the finiteness of the poset and the existence of maximal elements; this clarifies the contribution of the minimal-centralizer claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: [Proof that minimal centralizers cover the group] The argument that minimal centralizers cover G is less immediate than the maximal case. The manuscript should explicitly verify, for an arbitrary g, that the subposet of centralizers containing C(g) has a minimal element that is still a centralizer of some group element (rather than merely a minimal member of a larger collection).

Authors: We agree that the covering property for minimal centralizers benefits from a more explicit verification than the maximal case, which follows immediately from finiteness. In the revised manuscript we will add a short paragraph that, for arbitrary g, considers the subposet of all centralizers containing C(g). This subposet is nonempty (it contains C(g)) and finite, so it possesses minimal elements under inclusion. Any such minimal element is, by construction, the centralizer of some element of G and necessarily contains g. We will include this verification verbatim in the next version. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes that maximal centralizers under inclusion cover any finite group G, which follows immediately from the finiteness of the centralizer poset: every element g belongs to C(g), and the poset admits maximal elements containing it. The analogous claim for minimal centralizers is likewise a direct consequence of finiteness applied to the subposet of centralizers containing a fixed g. The congruence result is confined to the subclass of F-groups that are nonabelian p-groups and reduces to standard p-group counting once the definition of F-group is supplied; no fitted parameters, self-definitional equations, or load-bearing self-citations appear. The derivation relies on ordinary poset and group-theoretic facts that are independent of the paper's own results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard definition of the centralizer of an element and the partial order of subgroups by inclusion. These are ordinary background facts in group theory. No free parameters, ad-hoc axioms, or new postulated entities are visible in the abstract.

axioms (1)

standard math Standard axioms of group theory (associativity, identity element, inverses, closure).
The paper works inside the category of finite groups and therefore inherits the usual group axioms.

pith-pipeline@v0.9.0 · 5357 in / 1203 out tokens · 46242 ms · 2026-05-17T02:03:14.496251+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show that the set of centralizers that are maximal under the partial ordering form a cover of the group. We also show that the set of centralizers that are minimal under the partial ordering form a cover of the group.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A M\"obius function on the centralizer lattice
math.GR 2025-12 unverdicted novelty 5.0

A Möbius function is defined on the poset of element centers, producing new results about centralizers in p-groups.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

[1]

Bhargava, When is a group the union of proper normal subgroups?Amer

M. Bhargava, When is a group the union of proper normal subgroups?Amer. Math. Monthly 109(2002), 471-473

work page 2002
[2]

Brauer, M

R. Brauer, M. Suzuki, and G. E. Wall, A characterization of the one- dimensional unimodular projective groups over finite fields,Illinois Journal of Mathematics,2(1958), 718-745

work page 1958
[3]

Dolfi, M

S. Dolfi, M. Herzog and E. Jabara, Finite groups whose noncentral commuting elements have centralizers of equal size,Bull. Aust. Math. Soc.82(2010), 293-304

work page 2010
[4]

M. Garonzi. Finite Groups That Are the Union of at most 25 Proper Sub- groups,J. Algebra Appl.12(2013), 1350002

work page 2013
[5]

Garonzi, L.-C

M. Garonzi, L.-C. Kappe, and E. Swartz, On integers that are covering num- bers of groups,Exp. Math.31(2022), 425-443

work page 2022
[6]

Haji and S

S. Haji and S. M. J. Amiri, On groups covered by finitely many centralizers and domination number of the commuting graph,Comm. Algebra47(2019), 4641-4653

work page 2019
[7]

Itˆ o, On finite groups with given conjugate types I,Nagoya Math

N. Itˆ o, On finite groups with given conjugate types I,Nagoya Math. J.6(1953), 17-28

work page 1953
[8]

Kappe, Finite coverings: a journey through groups, loops, rings and semigroups, Group theory, combinatorics, and computing, 79–88,Contemp

L.-C. Kappe, Finite coverings: a journey through groups, loops, rings and semigroups, Group theory, combinatorics, and computing, 79–88,Contemp. Math.,611American Mathematical Society, Providence, RI, 2014 ISBN:978- 0-8218-9435-4

work page 2014
[9]

M. L. Lewis, A lower bound on the size of a maximal abelian subgroup, to appear in the Proceedings of Ischia Group Theory Conference 2024

work page 2024
[10]

M. L. Lewis and Ryan McCulloch, The commuting graph and a graph associated with centralizers, submitted for publication. https://doi.org/10.48550/arXiv.2511.11926

work page doi:10.48550/arxiv.2511.11926
[11]

D. R. Mason, On coverings of a finite group by abelian subgroups,Math. Proc. Cambridge Philos. Soc.83(1978), 205–209

work page 1978
[12]

Rebmann,F-Gruppen,Archiv

J. Rebmann,F-Gruppen,Archiv. Math. (Basel)22(1971), 225-230

work page 1971
[13]

Sun, Finite covers of groups by cosets or subgroups,Internat

Z.-W. Sun, Finite covers of groups by cosets or subgroups,Internat. J. Math. 17(2006), 1047-1064

work page 2006
[14]

Suzuki, The nonexistence of a certain type of simple groups of odd order, Proceedings of the American Mathematical Society,8(1957), 686-695

M. Suzuki, The nonexistence of a certain type of simple groups of odd order, Proceedings of the American Mathematical Society,8(1957), 686-695

work page 1957
[15]

Verardi, Gruppi semiextraspeciali di esponentep,Ann

L. Verardi, Gruppi semiextraspeciali di esponentep,Ann. Mat. Pura Appl. 148(1987) 131–171

work page 1987
[16]

Zappa, Partitions and other coverings of finite groups, Special issue in honor of Reinhold Baer (1902–1979),Illinois J

G. Zappa, Partitions and other coverings of finite groups, Special issue in honor of Reinhold Baer (1902–1979),Illinois J. Math.47(2003), 571-580

work page 1902
[17]

https://www2.math.binghamton.edu/p/zassenhaus/zassenhaus 2025/program 14 LEWIS AND MCCULLOCH Department of Mathematical Sciences, Kent State University, Kent, OH 44242; lewis@math.kent.edu Department of Mathematics and Statistics, Binghamton University, Binghamton, NY 13902; rmccullo1985@gmail.com

work page 2025

[1] [1]

Bhargava, When is a group the union of proper normal subgroups?Amer

M. Bhargava, When is a group the union of proper normal subgroups?Amer. Math. Monthly 109(2002), 471-473

work page 2002

[2] [2]

Brauer, M

R. Brauer, M. Suzuki, and G. E. Wall, A characterization of the one- dimensional unimodular projective groups over finite fields,Illinois Journal of Mathematics,2(1958), 718-745

work page 1958

[3] [3]

Dolfi, M

S. Dolfi, M. Herzog and E. Jabara, Finite groups whose noncentral commuting elements have centralizers of equal size,Bull. Aust. Math. Soc.82(2010), 293-304

work page 2010

[4] [4]

M. Garonzi. Finite Groups That Are the Union of at most 25 Proper Sub- groups,J. Algebra Appl.12(2013), 1350002

work page 2013

[5] [5]

Garonzi, L.-C

M. Garonzi, L.-C. Kappe, and E. Swartz, On integers that are covering num- bers of groups,Exp. Math.31(2022), 425-443

work page 2022

[6] [6]

Haji and S

S. Haji and S. M. J. Amiri, On groups covered by finitely many centralizers and domination number of the commuting graph,Comm. Algebra47(2019), 4641-4653

work page 2019

[7] [7]

Itˆ o, On finite groups with given conjugate types I,Nagoya Math

N. Itˆ o, On finite groups with given conjugate types I,Nagoya Math. J.6(1953), 17-28

work page 1953

[8] [8]

Kappe, Finite coverings: a journey through groups, loops, rings and semigroups, Group theory, combinatorics, and computing, 79–88,Contemp

L.-C. Kappe, Finite coverings: a journey through groups, loops, rings and semigroups, Group theory, combinatorics, and computing, 79–88,Contemp. Math.,611American Mathematical Society, Providence, RI, 2014 ISBN:978- 0-8218-9435-4

work page 2014

[9] [9]

M. L. Lewis, A lower bound on the size of a maximal abelian subgroup, to appear in the Proceedings of Ischia Group Theory Conference 2024

work page 2024

[10] [10]

M. L. Lewis and Ryan McCulloch, The commuting graph and a graph associated with centralizers, submitted for publication. https://doi.org/10.48550/arXiv.2511.11926

work page doi:10.48550/arxiv.2511.11926

[11] [11]

D. R. Mason, On coverings of a finite group by abelian subgroups,Math. Proc. Cambridge Philos. Soc.83(1978), 205–209

work page 1978

[12] [12]

Rebmann,F-Gruppen,Archiv

J. Rebmann,F-Gruppen,Archiv. Math. (Basel)22(1971), 225-230

work page 1971

[13] [13]

Sun, Finite covers of groups by cosets or subgroups,Internat

Z.-W. Sun, Finite covers of groups by cosets or subgroups,Internat. J. Math. 17(2006), 1047-1064

work page 2006

[14] [14]

Suzuki, The nonexistence of a certain type of simple groups of odd order, Proceedings of the American Mathematical Society,8(1957), 686-695

M. Suzuki, The nonexistence of a certain type of simple groups of odd order, Proceedings of the American Mathematical Society,8(1957), 686-695

work page 1957

[15] [15]

Verardi, Gruppi semiextraspeciali di esponentep,Ann

L. Verardi, Gruppi semiextraspeciali di esponentep,Ann. Mat. Pura Appl. 148(1987) 131–171

work page 1987

[16] [16]

Zappa, Partitions and other coverings of finite groups, Special issue in honor of Reinhold Baer (1902–1979),Illinois J

G. Zappa, Partitions and other coverings of finite groups, Special issue in honor of Reinhold Baer (1902–1979),Illinois J. Math.47(2003), 571-580

work page 1902

[17] [17]

https://www2.math.binghamton.edu/p/zassenhaus/zassenhaus 2025/program 14 LEWIS AND MCCULLOCH Department of Mathematical Sciences, Kent State University, Kent, OH 44242; lewis@math.kent.edu Department of Mathematics and Statistics, Binghamton University, Binghamton, NY 13902; rmccullo1985@gmail.com

work page 2025