A Torsion-free Supersoluble Group with Trivial Outer Automorphism Group

Ernesto Ingrosso; Marco Trombetti; Mattia Brescia

A Torsion-free Supersoluble Group with Trivial Outer Automorphism Group

T0 review · 0 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read There exists a torsion-free group of Hirsch length 14 with a normal series of infinite cyclic factors and no outer automorphisms.

desk verdict Explicit counter-example of Hirsch length 14 that settles Kourovka 13.23 at the exact ρ0=1 boundary left open by Menegazzo–Puglisi. read the letter →

arxiv 2607.11775 v1 pith:OMWQ4HQE submitted 2026-07-13 math.GR

Mattia Brescia , Ernesto Ingrosso , Marco Trombetti This is my paper · ORCID

classification math.GR MSC 20F2820F16

keywords outerautomorphismsupersolublegrouppolycyclicnormallypoly-ZKourovkaNotebooktorsion-freenilpotentHirschlength

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper answers a long-standing question in group theory by producing an explicit counterexample: a torsion-free group G of Hirsch length 14 that admits a finite chain of normal subgroups with every successive factor infinite cyclic, yet every automorphism of G is inner. Such groups are called normally poly-ℤ (or torsion-free supersoluble of a special kind). Earlier work had shown that any such group with abelianization rank at least 2 must have outer automorphisms, and that examples with trivial outer automorphism group can exist when the abelianization is finite; the remaining open case was precisely rank 1. The construction realises that boundary case: G is a semidirect product of a carefully chosen nilpotent group N of class 3 and Hirsch length 13 by an infinite cyclic group generated by an automorphism α of order 4 in Out(N). The resulting G has centre isomorphic to ℤ, abelianization ℤ⊕(ℤ/2ℤ)^{6}, and Out(G)=1. The existence of even one such group settles the Kourovka problem in the negative and shows that the rigidity phenomenon can occur for normally poly-ℤ groups.

What carries the argument

The nilpotent group N of class 3, Hirsch length 13, presented by generators X_i, Z_k, C subject to commutator relations read off a six-vertex graph, together with the explicit automorphism α that acts as -I on N_ab and satisfies α^{4} equal to conjugation by a fixed element R of N while α^{2} is not inner. The semidirect product G=⟨t⟩⋉_α N then has the desired normal series and Out(G)=1.

What would settle it

An explicit computation showing that the proposed images under α fail to preserve one of the commutator relations (N1)–(N3), or that α^{2} is in fact inner in N, or that some automorphism of the resulting semidirect product G is outer.

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Extended reading notes

Core claim

There exists a torsion-free group G of Hirsch length 14 that admits a finite normal series with every factor infinite cyclic and yet Out(G)=1. Moreover Z(G)≅ℤ and G_ab≅ℤ⊕(ℤ/2ℤ)^{6}. The group is realised as the semidirect product ⟨t⟩⋉_α N where N is a torsion-free nilpotent group of class 3 and Hirsch length 13 constructed from a rational Lie algebra associated with a six-vertex graph, and α is an automorphism of N of order exactly 4 in Out(N).

Load-bearing premise

The map α defined on the generators of N really extends to a group automorphism of N (checked only by verifying that the images satisfy the defining relations and that its fourth power is conjugation by an element of N).

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper closes Kourovka Problem 13.23 with a concrete torsion-free supersoluble group G of Hirsch length 14 that has a full normal series with infinite cyclic factors and Out(G)=1. That is the one fact that matters: previous work of Menegazzo–Puglisi handled ρ0≥2 and ρ0=0, and Robinson’s earlier examples were not normally poly-ℤ. The construction sits exactly on the narrow boundary they identified (Z(G)≅ℤ, Gab≅ℤ⊕(ℤ/2ℤ)6).

What they do well is keep everything explicit. N is built from a rational Lie algebra of class 3 attached to a six-vertex graph; the BCH formula gives a unique normal form, so N is torsion-free of Hirsch length 13. The automorphism α is written on generators, shown to preserve the relations by direct collection, and proved to satisfy α4=Inn(R-1) while α2 is not inner (the vector s lies outside the image of the incidence matrix D of det 2). The self-normalising property of ⟨[α]⟩≅C4 in Out(N) is a clean linear-algebra argument over ℤ and ℚ, and the final reduction that every automorphism of the semidirect product G=⟨t⟩⋉αN is inner follows the standard strategy without gaps. The stress-test note is right: the natural soft spot (whether α really extends) is already closed by the model.

Soft spots are minor and presentational. A couple of verifications (that α preserves every relation, uniqueness of the surjection G→ℤ) are left a little terse, and the Hirsch length 14 is not claimed to be minimal; the authors themselves ask what the minimal length is. None of this threatens the existence claim.

The paper is for people who work on outer automorphisms of infinite soluble or polycyclic groups. The calculations are fully displayed, so a careful reader can check them by hand. It deserves a serious referee and should be accepted after ordinary polishing. I would cite it when the topic comes up and would bring it to reading group if we are doing infinite groups this term.

Referee Report

0 major / 4 minor

Summary. The paper constructs a torsion-free supersoluble (normally poly-ℤ) group G of Hirsch length 14 with Out(G)=1, giving a negative answer to Kourovka Notebook Problem 13.23. The group is realized as the semidirect product G=⟨t⟩⋉_α N, where N is a torsion-free nilpotent group of class 3 and Hirsch length 13 built from a 6-vertex graph via a rational Lie algebra (Baker–Campbell–Hausdorff) model that yields a unique normal form. An explicit automorphism α of N is defined on generators so that [α] has order exactly 4 in Out(N) and is self-normalizing; adjoining t then forces every automorphism of G to be inner. The authors also record Z(G)≅ℤ and G_ab≅ℤ⊕(ℤ/2ℤ)^6, placing the example on the boundary left open by Menegazzo–Puglisi.

Significance. The result settles a 30-year-old Kourovka problem that had remained open precisely in the intermediate case ρ_0(G)=1. The construction is fully explicit (graph, Lie algebra, matrix D with det D=2, concrete formulas for α) and self-contained; the key linear-algebra verifications (order of [α], self-normalizer, reduction of Aut(G) to Inn(G)) are carried out by direct calculation rather than by appeal to general machinery. This supplies a concrete counter-example of modest Hirsch length and opens the natural minimal-length question posed at the end of the paper.

minor comments (4)

[Section 3] Section 3, definition of α: the verification that the six generator formulas preserve relations (N1)–(N3) is asserted by “collection”; a short expanded calculation (or a reference to a computer-algebra check) would make the step fully transparent for a reader who does not wish to recompute every commutator.
[Lemma 4.1] Lemma 4.1: the support restrictions on the columns of S are obtained by examining non-edges; listing the non-edges used (or giving a one-line matrix argument) would shorten the verification that S must be diagonal.
Throughout: a few typographical slips appear (missing spaces after periods, occasional “Wegive”-style concatenations in the abstract). A light copy-edit would improve readability.
[Question 5.3] Question 5.3: it would be useful to record the best lower bound currently known (e.g., from the Menegazzo–Puglisi constraints) so that the gap between 14 and the theoretical minimum is explicit.

Circularity Check

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No circularity: fully self-contained constructive existence proof with explicit generators, relations, and direct Out computations.

full rationale

The paper defines N via an explicit presentation (or equivalently a rational Lie algebra with BCH product) whose normal form and torsion-freeness are verified by direct logarithm and Jacobi checks; defines the endomorphism α by concrete formulas on the 13 generators and verifies it is an automorphism because the images satisfy (N1)–(N3) and α4 equals conjugation by R-1; forms the semidirect product G; then proves Out(G)=1 by a chain of explicit linear-algebra and conjugation calculations (det D=2, s not in im D, self-normalizing C4 in Out(N), centralizer of N, etc.). All steps are internal to the constructed objects; the few external citations (Robinson, Menegazzo–Puglisi, Kourovka, Segal) supply only background context or the problem statement and are not used as load-bearing uniqueness or ansatz inputs. No parameters are fitted, no quantity is predicted from a related fit, and no definitional loop appears. The derivation is therefore non-circular by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The paper is a pure existence proof inside finitely presented nilpotent and polycyclic groups. It relies only on standard Lie-algebra and collection arguments plus one ad-hoc choice of graph and automorphism formulas; no free parameters are fitted to data and no new physical or algebraic entities beyond the concrete group G are postulated.

assumptions (3)

standard math The Baker–Campbell–Hausdorff formula truncated at weight 3 defines a group law on the rational nilpotent Lie algebra n of class 3.
Used in Section 2 to realise N inside the simply-connected nilpotent Lie group; classical for nilpotent Lie algebras of class ≤3.
domain assumption A torsion-free supersoluble group with infinite abelianization and Aut=Inn must satisfy Z(G)≅Z, Z(G)∩G'=1 and ρ0(G)=1 (Menegazzo–Puglisi, Theorem 2).
Cited in the introduction to locate the counter-example on the unique remaining boundary case.
ad hoc to paper The six explicit formulas for α(Xi) and α(Zk) preserve the defining relations (N1)–(N3) of N.
Asserted after the definition of α in Section 3; the verification is left as a routine collection exercise.

invented entities (2)

The graph Γ on six vertices with the six listed edges, together with the associated nilpotent group N of class 3.
purpose: To furnish a concrete torsion-free nilpotent group of Hirsch length 13 admitting an outer automorphism of order 4 that can be used to build the semidirect product G.
The graph and the resulting commutator matrix D (det=2) are chosen by the authors so that s lies outside the image of D while 2s lies inside; no independent existence claim is made for the graph outside this construction.
The automorphism α of N (and the resulting semidirect product G=⟨t⟩⋉_α N).
purpose: To produce a normally poly-ℤ group whose outer automorphism group is forced to be trivial.
Defined by explicit action on generators; its order-4 class in Out(N) is the key new algebraic object.

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Cite this review

Pith. "Pith review of A Torsion-free Supersoluble Group with Trivial Outer Automorphism Group." pith.science (2026). https://pith.science/paper/OMWQ4HQE

@misc{pith2026260711775,
  author       = {Pith},
  title        = {Pith review of: A Torsion-free Supersoluble Group with Trivial Outer Automorphism Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OMWQ4HQE}},
  note         = {Machine review of arXiv:2607.11775}
}

abstract

We give a negative solution to Problem~13.23 of the Kourovka Notebook. We construct a torsion-free group $G$ of Hirsch length $14$ admitting a finite series \[ 1=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_{14}=G \] in which every $G_i$ is normal in $G$ and every factor is infinite cyclic, but such that $\Out(G)=1$.

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Reference graph

Works this paper leans on

5 extracted references · 1 linked inside Pith

[1]

S. G. Dani and M. G. Mainkar,Anosov automorphisms on compact nilmanifolds associated with graphs, Trans. Amer. Math. Soc.357(2005), 2235–2251

2005
[2]

E. I. Khukhro and V. D. Mazurov (eds.),Unsolved Problems in Group Theory. The Kourovka Notebook, No. 21, Sobolev Institute of Mathematics, Novosibirsk, 2026; Problem 13.23, proposed by F. de Giovanni. Available as arXiv:1401.0300v45

arXiv 2026
[3]

Menegazzo and O

F. Menegazzo and O. Puglisi,Outer automorphisms of supersoluble groups, Glasgow Math. J.42(2000), 115–120

2000
[4]

D. J. S. Robinson,Infinite soluble groups with no outer automorphisms, Rend. Sem. Mat. Univ. Padova62(1980), 281–294

1980
[5]

Renato Caccioppoli

D. Segal,Polycyclic Groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. Mattia Brescia, Ernesto Ingrosso, Marco Trombetti Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” Università di Napoli Federico II Complesso Universitario Monte S. Angelo Via Cintia, Napoli (Italy) e-mail: mattia.brescia@unina....

1983

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.

[1] [1]

S. G. Dani and M. G. Mainkar,Anosov automorphisms on compact nilmanifolds associated with graphs, Trans. Amer. Math. Soc.357(2005), 2235–2251

2005

[2] [2]

E. I. Khukhro and V. D. Mazurov (eds.),Unsolved Problems in Group Theory. The Kourovka Notebook, No. 21, Sobolev Institute of Mathematics, Novosibirsk, 2026; Problem 13.23, proposed by F. de Giovanni. Available as arXiv:1401.0300v45

arXiv 2026

[3] [3]

Menegazzo and O

F. Menegazzo and O. Puglisi,Outer automorphisms of supersoluble groups, Glasgow Math. J.42(2000), 115–120

2000

[4] [4]

D. J. S. Robinson,Infinite soluble groups with no outer automorphisms, Rend. Sem. Mat. Univ. Padova62(1980), 281–294

1980

[5] [5]

Renato Caccioppoli

D. Segal,Polycyclic Groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. Mattia Brescia, Ernesto Ingrosso, Marco Trombetti Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” Università di Napoli Federico II Complesso Universitario Monte S. Angelo Via Cintia, Napoli (Italy) e-mail: mattia.brescia@unina....

1983