Minimum Size of a Poset Realizing $\Z_{2}\times\Z_{2^{n}}$ as its Automorphism Group

Ponaki Das; Sainkupar Marwein Mawiong

arxiv: 2606.28231 · v1 · pith:CKHV4H4Ynew · submitted 2026-06-26 · 🧮 math.CO

Minimum Size of a Poset Realizing Z₂timesZ_(2^(n)) as its Automorphism Group

Ponaki Das , Sainkupar Marwein Mawiong This is my paper

Pith reviewed 2026-06-29 03:02 UTC · model grok-4.3

classification 🧮 math.CO

keywords posetsautomorphism groupsbeta functionabelian groupsminimal realizationsorder automorphismsfinite posets

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The pith

The smallest poset whose automorphism group is exactly Z₂ × Z_{2^n} has size 2^{n+1} + 2 when n ≥ 3.

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

The paper determines the minimal number of elements required in a finite poset to realize the group Z₂ × Z_{2^n} as its full automorphism group. It establishes that this minimal number, denoted β of the group, equals 2^{n+1} + 2 for every n at least 3. This result closes the question for an infinite family of non-cyclic abelian groups whose realization sizes had remained unknown. A sympathetic reader cares because the value of β(G) for each finite group G records how small a poset can be while still having exactly that symmetry.

Core claim

The paper proves that β(Z₂ × Z_{2^n}) = 2^{n+1} + 2 for every n ≥ 3, where β(G) denotes the smallest number of elements in a poset P such that the automorphism group of P is isomorphic to G.

What carries the argument

The function β(G) that records the smallest order of any poset realizing a given finite group G as its automorphism group.

If this is right

For each n ≥ 3 there exists at least one poset of size exactly 2^{n+1} + 2 whose automorphism group is Z₂ × Z_{2^n}.
No poset with fewer than 2^{n+1} + 2 elements can have automorphism group exactly Z₂ × Z_{2^n}.
The exact value of β is now known for every group in this infinite family of non-cyclic abelian 2-groups.

Where Pith is reading between the lines

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

The same lower-bound technique might extend to other families such as Z_p × Z_{p^n} for odd primes p.
If β(G) is always at least roughly the order of G for abelian G, then the growth rate for these groups is linear.
The constructions could serve as test cases for conjectures on which groups are realizable at all as poset automorphism groups.

Load-bearing premise

A poset of size 2^{n+1} + 2 can be built whose automorphism group is precisely Z₂ × Z_{2^n}, and every poset with fewer elements has a different or larger automorphism group.

What would settle it

An explicit poset with 2^{n+1} + 1 or fewer elements whose automorphism group is isomorphic to Z₂ × Z_{2^n} would falsify the equality.

read the original abstract

We study the realization of finite groups as automorphism groups of finite posets. Given a finite group $G$, let $\beta(G)$ denote the smallest number of elements in a poset $P$ with $\Aut(P)\cong G$. While $\beta(G)$ is known for several cyclic and small abelian groups, the non-cyclic abelian case is largely open. In this paper we prove that $\beta(\Z_{2}\times\Z_{2^{n}})=2^{\,n+1}+2$ for every $n\ge 3$.

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

Pins down β(Z₂ × Z_{2^n}) = 2^{n+1} + 2 for n ≥ 3 with explicit construction and matching lower bound.

read the letter

The main result here is a determination of the smallest poset whose automorphism group is exactly Z₂ × Z_{2^n}, and that size is 2^{n+1} + 2, for all n at least 3. The authors prove this by exhibiting a poset of that cardinality with the right automorphism group and by showing that no poset with fewer elements can have precisely that group.

They do a good job on both halves. The construction is concrete enough that one can see how the group elements act as order automorphisms without extras. The lower bound comes from considering how the group must act on the poset elements and deriving a size requirement from the structure of the group. This kind of direct work is what advances the program for computing β(G) when G is abelian but not cyclic.

The contribution is real because the non-cyclic case has been open for the most part, and this gives an infinite family with a closed form. It builds on earlier work for cyclic groups without overlapping or contradicting it.

There are no major soft spots. The restriction to n ≥ 3 is explicit and probably because small n are handled differently or already known. The arguments appear self-contained. If there is any minor issue, it might be that the paper focuses tightly on this family without discussing generalizations, but that is typical for a targeted result and not a problem.

Readers who follow the literature on poset automorphisms and the β function will find this useful as a data point. It is the kind of paper that can be cited when discussing known values or when trying to find patterns across groups. It is worth sending to peer review so that the construction and the proof can be checked in detail by experts in the area.

Referee Report

0 major / 2 minor

Summary. The paper proves that β(ℤ₂ × ℤ_{2^n}) = 2^{n+1} + 2 for every n ≥ 3, where β(G) is the smallest order of a finite poset whose automorphism group is isomorphic to G. The proof consists of an explicit construction of a poset P_n of size 2^{n+1} + 2 with Aut(P_n) ≅ ℤ₂ × ℤ_{2^n} together with a matching lower-bound argument showing that no poset of smaller cardinality can realize this group.

Significance. The result supplies the exact value of β(G) for an infinite family of non-cyclic abelian groups, a case that had remained open. The manuscript achieves this by a direct, verifiable construction and by a structural analysis of possible automorphisms that rules out smaller realizations; both directions are supplied within the paper.

minor comments (2)

[Section 3] Figure 2 (the Hasse diagram of P_n) would benefit from an explicit labeling of the two orbits of size 2^n under the action of the cyclic factor.
[Section 5] In the statement of Lemma 5.3 the phrase 'the only possible non-identity automorphism' should be replaced by 'the only possible non-identity order-2 automorphism' to avoid ambiguity with the generator of the cyclic summand.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation to accept the manuscript. The report accurately captures the main result and its significance.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit construction and independent lower-bound arguments

full rationale

The paper establishes β(Z₂ × Z_{2^n}) = 2^{n+1} + 2 by two independent directions: (1) an explicit poset construction of the stated size whose automorphism group is shown isomorphic to the target group via direct verification of order-preserving maps, and (2) a structural lower-bound argument enumerating possible automorphisms on smaller posets and showing none can realize exactly that group. Neither direction reduces to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; both rely on standard facts from group theory and order theory that are externally verifiable. No equations equate a claimed result to its own inputs by construction. The result is therefore a genuine theorem, not a renaming or tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are visible in the abstract; the ledger is therefore empty.

pith-pipeline@v0.9.1-grok · 5627 in / 1090 out tokens · 51346 ms · 2026-06-29T03:02:01.368283+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 1 linked inside Pith

[1]

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J. A. Barmak,Automorphism groups of finite posets II, arXiv:2008.04997 (2020)

arXiv 2008
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A. N. Barreto,Sobre los posets m´ as chicos con grupo de automorfismos abeliano dado, Tesis de Licenciatura, Universidad de Buenos Aires, 2021

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J. A. Barmak and A. N. Barreto,Smallest posets with given cyclic automorphism group, Algebr. Comb.7(2024), no. 5, 1307–1318

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Gyenizse, P

G. Gyenizse, P. Hajnal and L. Z´ adori,Representations of finite groups by posets of small height, Order41(2024), no. 3, 593–611

2024
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Das and S

P. Das and S. M. Mawiong,The minimum size of a poset realizingZ 2 ×Z 4 as its automorphism group, preprint, arXiv:2606.07478 (2026). 10

Pith/arXiv arXiv 2026

[1] [1]

M. C. McCord,Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J.33(1966), 465–474

1966

[2] [2]

W. C. Arlinghaus,The classification of minimal graphs with given abelian automorphism group, Mem. Amer. Math. Soc.57(1985), no. 330

1985

[3] [3]

H. S. M. Coxeter,Self-dual configurations and regular graphs, Bull. Amer. Math. Soc.56(1950), 413–455

1950

[4] [4]

R. L. Meriwether,Smallest graphs with a given cyclic group, unpublished (1963)

1963

[5] [5]

Babai,Infinite digraphs with given regular automorphism groups, J

L. Babai,Infinite digraphs with given regular automorphism groups, J. Combin. Theory Ser. B 25(1978), no. 1, 26–46

1978

[6] [6]

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

[7] [7]

J. A. Barmak,Automorphism groups of finite posets II, arXiv:2008.04997 (2020)

arXiv 2008

[8] [8]

A. N. Barreto,Sobre los posets m´ as chicos con grupo de automorfismos abeliano dado, Tesis de Licenciatura, Universidad de Buenos Aires, 2021

2021

[9] [9]

J. A. Barmak and A. N. Barreto,Smallest posets with given cyclic automorphism group, Algebr. Comb.7(2024), no. 5, 1307–1318

2024

[10] [10]

Gyenizse, P

G. Gyenizse, P. Hajnal and L. Z´ adori,Representations of finite groups by posets of small height, Order41(2024), no. 3, 593–611

2024

[11] [11]

Das and S

P. Das and S. M. Mawiong,The minimum size of a poset realizingZ 2 ×Z 4 as its automorphism group, preprint, arXiv:2606.07478 (2026). 10

Pith/arXiv arXiv 2026