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arxiv: 2604.20501 · v1 · submitted 2026-04-22 · 🧮 math.LO

An unusual example of a universal automorphism group

Rob Sullivan , Jeroen Winkel This is my paper

Pith reviewed 2026-05-09 22:50 UTC · model grok-4.3

classification 🧮 math.LO
keywords Fraïssé structureuniversal automorphism groupgroup-extensible ω-ageultrahomogeneous structureω-agemodel theorypermutation group
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The pith

A Fraïssé structure admits embeddings of all automorphism groups from its embeddable substructures yet fails the stronger condition that those automorphisms extend while preserving group composition.

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

The paper builds a countably infinite ultrahomogeneous structure M whose automorphism group receives an embedding from Aut(A) for every structure A that embeds into M. For at least one such A, however, no copy of A inside M has the property that every automorphism of the copy extends to an automorphism of M in a way that respects the group operation. This separates two notions that stand in one-way implication: group-extensible ω-age immediately yields a universal automorphism group, but the converse does not hold. The example therefore shows that universality of the automorphism group is strictly weaker than the extensibility condition on the age.

Core claim

We give an example of a Fraïssé structure with a universal automorphism group whose ω-age is not group-extensible, showing that the above two properties are not equivalent.

What carries the argument

The specific Fraïssé structure M constructed in the paper, which is ultrahomogeneous and countably infinite, and whose ω-age consists of all structures embeddable into M; this M allows Aut(A) to embed into Aut(M) for every such A, yet for some A no embedding of A into M makes the automorphism extensions respect group composition.

Load-bearing premise

The structure constructed in the paper really is ultrahomogeneous with the stated age and really satisfies the universal-embedding property for automorphism groups while failing the extensibility property for at least one substructure.

What would settle it

A direct check that every structure embeddable in M admits an embedding into M whose automorphisms all extend to M while preserving the group law would falsify the claimed separation.

read the original abstract

Let $M$ be a Fra\"{i}ss\'{e} structure (a countably infinite ultrahomogeneous structure). We refer to the class of structures embeddable in $M$ as the $\omega$-age of $M$. We consider the following two properties of $M$: we say that $M$ has a universal automorphism group if, for each $A$ in the $\omega$-age of $M$, there is an embedding $\textrm{Aut}(A) \to \textrm{Aut}(M)$, and we say that $M$ has group-extensible $\omega$-age if, for each $A$ in the $\omega$-age of $M$, there is an embedding $A \to M$ such that each automorphism of the image extends to an automorphism of $M$ and the extension map preserves group composition. It is immediate that if $M$ has group-extensible $\omega$-age, then $M$ has a universal automorphism group. We give an example of a Fra\"{i}ss\'{e} structure with a universal automorphism group whose $\omega$-age is not group-extensible, showing that the above two properties are not equivalent.

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 constructs a specific Fraïssé structure M (ultrahomogeneous countable structure) whose ω-age admits embeddings Aut(A) ↪ Aut(M) for every finite A in the age, but for which there exists some A where no embedding A ↪ M has the property that every automorphism of the image extends to an automorphism of M while preserving composition. The construction uses a relational language with carefully chosen forbidden substructures to ensure ultrahomogeneity (via explicit back-and-forth) while blocking group-extensibility for one particular A (via case analysis on possible images).

Significance. The result separates the notions of universal automorphism group and group-extensible ω-age for Fraïssé structures, which had not previously been shown to be inequivalent. The explicit construction, including the back-and-forth verification of ultrahomogeneity and the case-by-case obstruction to extendability, is a strength that makes the counterexample verifiable and potentially useful for further work on automorphism groups of homogeneous structures.

minor comments (2)
  1. [§2] §2 (Construction): the definition of the age via forbidden configurations is clear, but a short table listing the allowed finite structures up to isomorphism would improve readability for readers checking the back-and-forth argument.
  2. [§4] §4 (Non-extensibility): the case analysis on images of the critical A is complete, but the notation distinguishing automorphisms of the image from their potential extensions could be made more uniform to avoid minor confusion in the contradiction step.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in separating the two notions, and recommendation to accept. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; explicit counterexample construction

full rationale

The paper constructs a specific Fraïssé structure M and verifies its properties through direct, self-contained arguments: ultrahomogeneity via an explicit back-and-forth argument extending isomorphisms between finite substructures; the universal automorphism group property via explicit abstract embeddings Aut(A) ↪ Aut(M) for each finite A in the age; and failure of group-extensibility via case analysis on possible images of a particular A, using forbidden configurations in the age to derive contradictions. These steps do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The implication that group-extensible ω-age implies universal automorphism group is stated as immediate and is not part of the main claim. The derivation chain is independent of the target result and consists of standard model-theoretic constructions and verifications.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard set-theoretic and model-theoretic background to assert existence of a countable ultrahomogeneous structure with the stated properties; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math ZFC set theory and the definition of Fraïssé structures (countable ultrahomogeneous relational structures with the amalgamation property).
    Invoked to guarantee the existence of the structure M and its age.

pith-pipeline@v0.9.0 · 5508 in / 1084 out tokens · 24240 ms · 2026-05-09T22:50:36.774701+00:00 · methodology

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Aranda, A. and. Ramsey expansions of metrically homogeneous graphs , NOTE =

    work page

  2. [2]

    Archive for Mathematical Logic , volume=

    On countable homogeneous 3-hypergraphs , author=. Archive for Mathematical Logic , volume=. 1995 , publisher=

    work page 1995

  3. [3]

    The Journal of Symbolic Logic , volume=

    Free amalgamation and automorphism groups , author=. The Journal of Symbolic Logic , volume=. 2016 , publisher=

    work page 2016

  4. [4]

    2012 , school=

    Automorphism groups of homogeneous structures , author=. 2012 , school=

    work page 2012

  5. [5]

    Contributions to Discrete Mathematics , volume=

    Some rigid moieties of homogeneous graphs , author=. Contributions to Discrete Mathematics , volume=

    work page

  6. [6]

    Contributions to Discrete Mathematics , volume=

    Elements of finite order in automorphism groups of homogeneous structures , author=. Contributions to Discrete Mathematics , volume=

    work page

  7. [7]

    , author=

    Ramsey classes: examples and constructions. , author=. Surveys in Combinatorics , volume=

    work page

  8. [8]

    The linear isometry group of the

    Ben Yaacov, Ita. The linear isometry group of the. Proceedings of the American Mathematical Society , volume=

    work page

  9. [9]

    Mathematische Zeitschrift , volume=

    Transitivity of permutation groups on unordered sets , author=. Mathematische Zeitschrift , volume=. 1976 , publisher=

    work page 1976

  10. [10]

    Cameron, Peter J. , year=. Oligomorphic Permutation Groups , publisher=

    work page

  11. [11]

    1987 , journal =

    Homogeneous directed graphs: the imprimitive case , series =. 1987 , journal =

    work page 1987

  12. [12]

    , TITLE =

    Cherlin, Gregory L. , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 1998 , NUMBER =

    work page 1998

  13. [13]

    Journal of Algebra , volume=

    Simplicity of the automorphism groups of order and tournament expansions of homogeneous structures , author=. Journal of Algebra , volume=. 2021 , publisher=

    work page 2021

  14. [14]

    Journal of Algebra , volume=

    Simplicity of the automorphism groups of generalised metric spaces , author=. Journal of Algebra , volume=. 2021 , publisher=

    work page 2021

  15. [15]

    Folkman, Jon , TITLE =. SIAM J. Appl. Math. , FJOURNAL =. 1970 , PAGES =

    work page 1970

  16. [16]

    Symmetrized induced R amsey theory

    Geschke, Stefan and Kojman, Menachem. Symmetrized induced R amsey theory. Graphs and Combinatorics. 2011

    work page 2011

  17. [17]

    An example of a

    Greb. An example of a. Applied Categorical Structures , volume=. 2018 , publisher=

    work page 2018

  18. [18]

    Pacific Journal of Mathematics , volume=

    A family of countable homogeneous graphs , author=. Pacific Journal of Mathematics , volume=. 1971 , publisher=

    work page 1971

  19. [19]

    Model Theoretic Methods in Finite Combinatorics , year =

    Hasson, Assaf and Kojman, Menachem and Onshuus, Alf , title =. Model Theoretic Methods in Finite Combinatorics , year =

    work page

  20. [20]

    Hodges, Wilfrid , TITLE =

    work page

  21. [21]

    Naturality and definability

    Hodges, Wilfrid and Shelah, Saharon , year=. Naturality and definability. Cubo (Temuco) , volume=

    work page

  22. [22]

    Combinatorica , volume=

    On stabilizers of some moieties of the random tournament , author=. Combinatorica , volume=. 2007 , publisher=

    work page 2007

  23. [23]

    On universal metric spaces , BOOKTITLE =

    Kat. On universal metric spaces , BOOKTITLE =. 1988 , PUBLISHER =

    work page 1988

  24. [24]

    Kubi. Kat. Applied Categorical Structures , volume=. 2017 , publisher=

    work page 2017

  25. [25]

    Coloring of universal graphs , JOURNAL =

    Komj\'. Coloring of universal graphs , JOURNAL =. 1986 , NUMBER =

    work page 1986

  26. [26]

    Automorphisms of the

    Kaplan, Itay and Riahi, Binyamin and. Automorphisms of the. Journal of Algebra , volume=. 2025 , publisher=

    work page 2025

  27. [27]

    Transactions of the American Mathematical Society , volume=

    Automorphism groups of finite topological rank , author=. Transactions of the American Mathematical Society , volume=

    work page

  28. [28]

    Journal of Symbolic Logic , volume=

    Homogeneous structures with nonuniversal automorphism groups , author=. Journal of Symbolic Logic , volume=

    work page

  29. [29]

    Extensive embeddings into

    Kwiatkowska, Aleksandra and Sullivan, Rob and Winkel, Jeroen , journal=. Extensive embeddings into

    work page

  30. [30]

    , TITLE =

    Lachlan, Alistair H. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1984 , NUMBER =

    work page 1984

  31. [31]

    Automorphism groups of homogeneous structures with stationary weak independence relations , author=

    work page

  32. [32]

    Automorphism groups of linearly ordered homogeneous structures , author=

    work page

  33. [33]

    2021 , school =

    Li, Yibei , title =. 2021 , school =

    work page 2021

  34. [34]

    Discrete Mathematics , volume=

    A survey of homogeneous structures , author=. Discrete Mathematics , volume=. 2011 , publisher=

    work page 2011

  35. [35]

    The Journal of Symbolic Logic , VOLUME =

    Meir, Nadav , TITLE =. The Journal of Symbolic Logic , VOLUME =. 2016 , NUMBER =

    work page 2016

  36. [36]

    Stabilizers of closed sets in the

    Melleray, Julien , journal=. Stabilizers of closed sets in the. 2006 , volume =

    work page 2006

  37. [37]

    Journal of Algebra , volume=

    Simplicity of some automorphism groups , author=. Journal of Algebra , volume=. 2011 , publisher=

    work page 2011

  38. [38]

    M. Fra\". Journal of Algebra , VOLUME =. 2016 , PAGES =

    work page 2016

  39. [39]

    Forum Mathematicum , year=

    The permutation group induced on a moiety , author=. Forum Mathematicum , year=

    work page

  40. [40]

    arXiv preprint arXiv:1703.10517 , year=

    The strong small index property for free homogeneous structures , author=. arXiv preprint arXiv:1703.10517 , year=

    work page arXiv

  41. [41]

    2013 , publisher=

    Ramsey theory for discrete structures , author=. 2013 , publisher=

    work page 2013

  42. [42]

    Ergodic Theory and Dynamical Systems , volume=

    Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs , author=. Ergodic Theory and Dynamical Systems , volume=. 2020 , publisher=

    work page 2020

  43. [43]

    Geometry and Combinatorics , pages=

    A survey of two-graphs , author=. Geometry and Combinatorics , pages=. 1991 , publisher=

    work page 1991

  44. [44]

    On the isometry group of the

    Tent, Katrin and Ziegler, Martin , journal=. On the isometry group of the. 2013 , publisher=

    work page 2013

  45. [45]

    On the group of isometries of the

    Uspenskij, Vladimir Vladimirovich , journal=. On the group of isometries of the. 1990 , publisher=

    work page 1990