arxiv: 2605.03449 · v1 · submitted 2026-05-05 · 🧮 math.GR · math.CO

Recognition: unknown

On the base size of a finite group on its action on the lattice of subgroups

JiaLi Du, Joy Morris, Pablo Spiga

Pith reviewed 2026-05-09 16:29 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords base sizeautomorphism groupsubgroup latticefinite groupscyclic groupspermutation action

0 comments

The pith

The base size of Aut(R) acting on the subgroup lattice of finite group R equals 1 exactly when R is cyclic.

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 any finite group R the smallest collection of subgroups with trivial common stabilizer under Aut(R) has size one if and only if R is cyclic. This tells us precisely when a single subgroup is enough to force any automorphism that preserves it to be the identity. The result matters for questions about recovering a group's automorphism group from limited data on its subgroup structure, especially in the context of representing arbitrary groups as automorphism groups of lattices.

Core claim

The base size of the natural action of Aut(R) on the lattice of all subgroups of R is equal to 1 if and only if R is cyclic.

What carries the argument

Base size of a group action: the smallest number of points (here, subgroups) whose pointwise stabilizer in Aut(R) is the identity.

If this is right

When R is cyclic there exists a subgroup fixed solely by the identity automorphism.
When R is non-cyclic every individual subgroup admits a non-identity automorphism that preserves it, so at least two subgroups are required.
The subgroup lattice of R distinguishes the full automorphism group with a single point precisely for cyclic groups.
The classification supplies an exact criterion for when the natural action on subgroups has an orbit whose size equals |Aut(R)|.

Where Pith is reading between the lines

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

Similar base-size calculations could be performed for the action on other natural lattices attached to groups, such as the lattice of normal subgroups.
The result separates cyclic groups as those for which the subgroup lattice carries enough information to recover Aut(R) from one preserved element.
One could test whether the same cyclic/non-cyclic threshold appears when the action is restricted to characteristic subgroups or to subgroups of fixed order.

Load-bearing premise

R is finite and the action is the natural one of Aut(R) on the full set of subgroups of R.

What would settle it

Either a cyclic finite group in which every subgroup is fixed by some non-identity automorphism, or a non-cyclic finite group possessing at least one subgroup fixed only by the identity automorphism.

read the original abstract

Given a finite group $R$, we investigate the base size of the action of the automorphism group of $R$ on the lattice of subgroups of $R$. Our main result shows that this base size is $1$ if and only if $R$ is cyclic. Our motivation arises from a conjecture of Babai on the problem of representing groups as automorphism groups of lattices with a bounded number of orbits.

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

1 major / 1 minor

Summary. The manuscript investigates the base size of the natural action of Aut(R) on the subgroup lattice L(R) for a finite group R. The central claim is that this base size equals 1 if and only if R is cyclic, motivated by Babai's conjecture on representing groups as automorphism groups of lattices with bounded orbits.

Significance. If the result held, it would classify precisely when a single subgroup determines Aut(R) via its pointwise stabilizer in L(R), offering a clean dichotomy with potential applications to orbit bounds in lattice representations of groups and to Babai's conjecture.

major comments (1)

[Abstract / Main Result] Abstract / Main Result: The assertion that base size equals 1 for cyclic R is false. When R is cyclic, every subgroup is characteristic (unique of given order), so Aut(R) acts trivially on L(R). For any collection of subgroups the pointwise stabilizer is then the whole Aut(R). Thus, whenever |Aut(R)| > 1 (e.g., R = ℤ/3ℤ), no base of size 1 exists, contradicting the claimed equivalence.

minor comments (1)

The abstract would benefit from a short reminder of the definition of base size (minimal size of a set with trivial pointwise stabilizer) for readers outside permutation-group theory.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying an error in the statement of the main result. We acknowledge that the claimed equivalence does not hold, and we will revise the paper to correct the theorem, the abstract, and any related discussion. We appreciate the referee's context on potential applications to Babai's conjecture.

read point-by-point responses

Referee: The assertion that base size equals 1 for cyclic R is false. When R is cyclic, every subgroup is characteristic (unique of given order), so Aut(R) acts trivially on L(R). For any collection of subgroups the pointwise stabilizer is then the whole Aut(R). Thus, whenever |Aut(R)| > 1 (e.g., R = ℤ/3ℤ), no base of size 1 exists, contradicting the claimed equivalence.

Authors: We agree with the referee's analysis. For cyclic R, all subgroups are indeed characteristic, so Aut(R) fixes every element of L(R) and the action is trivial. The pointwise stabilizer of any set of subgroups is therefore the full Aut(R). When |Aut(R)| > 1, no base of size 1 can exist. This shows that the 'if' direction of our main theorem is incorrect. We will revise the statement of the main result to classify the base size correctly: it equals 1 only in the cases where Aut(R) acts faithfully with a regular orbit or similar (specifically, only for the trivial group and ℤ/2ℤ among cyclic groups). For all other cyclic groups the base size is undefined in the standard sense or greater than 1. We will also verify and, if necessary, strengthen the argument for non-cyclic groups showing that the base size exceeds 1, and we will update the abstract and motivation section accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a direct mathematical proof of its main theorem on base sizes in group actions, with no visible fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the claimed result to its own inputs by construction. The derivation relies on standard group-theoretic arguments about automorphism actions on subgroup lattices and is self-contained against external benchmarks such as the definitions of base size and characteristic subgroups. No equations or reductions matching the enumerated circularity patterns are present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned.

pith-pipeline@v0.9.0 · 5356 in / 912 out tokens · 26498 ms · 2026-05-09T16:29:25.609497+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

13 extracted references

[1]

Babai,Finite digraphs with given regular automorphism groups, Period

L. Babai,Finite digraphs with given regular automorphism groups, Period. Math. Hungar. 11(1980), no. 4, 257–270

1980
[2]

Babai,On the abstract group of automorphisms, Combinatorics, Proceedings 8th British Combinatorics Conference, Swansea 1981, ed

L. Babai,On the abstract group of automorphisms, Combinatorics, Proceedings 8th British Combinatorics Conference, Swansea 1981, ed. H. N. V. Temperley; London Mathematical Society Lecture Notes 52 (University Press, Cambridge, 1981) 1–40

1981
[3]

Babai, A

L. Babai, A. J. Goodman,Subdirectly reducible groups and edge-minimal graphs with given automorphism group, J. London Math. Soc.47(1993), 417–432

1993
[4]

Babai, A

L. Babai, A. J. Goodman, L. Lov´ asz,Graphs with given automorphism group and few edge orbits, European J. Combin.12(1991), 185–203

1991
[5]

Baer,Der Kern, eine charakteristische Untergruppe, Compositio Math.1(1935), 254–283

R. Baer,Der Kern, eine charakteristische Untergruppe, Compositio Math.1(1935), 254–283

1935
[6]

Birkhoff,Sobre los grupos de automorfismos, Rev

G. Birkhoff,Sobre los grupos de automorfismos, Rev. Un. Mat. Argentina11(1946), 155–157

1946
[7]

T. C. Burness, M. Garonzi, A. Lucchini,Finite groups, minimal bases and the intersection number, Trans. Amer. Math. Soc.9(2022), 20–55

2022
[8]

T. C. Burness, H. Y. Huang,On the intersection of Sylow subgroups in almost simple groups, J. Algebra690(2026), 596–631

2026
[9]

Dedekind, ¨Uber Gruppen, deren s¨ ammtliche Theiler Normaltheiler sind, Mathematische Annalen48(1897), 548–561

R. Dedekind, ¨Uber Gruppen, deren s¨ ammtliche Theiler Normaltheiler sind, Mathematische Annalen48(1897), 548–561
[10]

Frucht,Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math

R. Frucht,Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math. 6(1938), 239–250

1938
[11]

A. J. Goodman,The edge-orbit conjecture of Babai, J. Combin. Theory Ser. B57(1993), 26–35

1993
[12]

Malle, T

G. Malle, T. Weigel,Generation of classical groups, Geometriae Dedicata49(1995), 85–116. BASE SIZE 11

1995
[13]

Schenkman,On the norm of a group, Illinois J

E. Schenkman,On the norm of a group, Illinois J. Math.4(1960), 150–152. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, P.R. China, Ministry of Education Key Laboratory of NSLSCS, Nanjing, 210023, P.R. China Email address:dujl@njnu.edu.cn Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB....

1960