[#P2858] Ternary representatives of finite-order integral matrices

A mathematical schematic of Ternary representatives of finite-order integral matrices. — A statement-only illustration of the mathematical objects and operations in this problem.

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Problem. Is every finite-order matrix \(A\in\operatorname{GL}_n(\mathbb Z)\), for every \(n\ge1\), conjugate within \(\operatorname{GL}_n(\mathbb Z)\) to a matrix whose entries all lie in \(\{-1,0,1\}\)? If the answer is negative, determine the least dimension containing a counterexample.

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Definitions and notation

1Context

Each fixed dimension has finitely many integral conjugacy classes of torsion elements. Class representatives, lattice invariants, and conjugacy certificates form a cumulative search corpus.

2Definitions

Definition 1. Finite order means that \(A^r=I_n\) for some positive integer \(r\).

Definition 2. Integral conjugacy means replacing \(A\) by \(PAP^{-1}\) for some \(P\in\operatorname{GL}_n(\mathbb Z)\).

Definition 3. A ternary matrix is an integer matrix all of whose entries belong to \(\{-1,0,1\}\).

3What counts as a solution

For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative.
For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions.

1Status

What counts as a solution

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The source thread verifies the property through dimension four using classifications of integral torsion conjugacy classes. The dated search found no general theorem or higher-dimensional counterexample. Exact unresolved remainder: For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions.^[2][1][3]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: The source thread verifies the property through dimension four using classifications of integral torsion conjugacy classes. The dated search found no general theorem or higher-dimensional counterexample. Exact unresolved remainder: For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The source thread verifies the property through dimension four using classifications of integral torsion conjugacy classes. The dated search found no general theorem or higher-dimensional counterexample. Exact unresolved remainder: For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions.^[2][1][3] Relevance to this problem For Ternary representatives of finite-order integral matrices, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1, Reference 3 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The source thread verifies the property through dimension four using classifications of integral torsion conjugacy classes. The dated search found no general theorem or higher-dimensional counterexample. Exact unresolved remainder: For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions. Open full record →
For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions.^[1] Relevance to this problem For finite order integer matrix ternary conjugate, sets the next completion test: For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. For a positive answer, prove that every finite-order element of every \(\operatorname{GL}_n(\mathbb Z)\) has an integrally conjugate ternary representative. For a negative answer, give a finite-order integral matrix in the least possible dimension and prove that no integral conjugate is ternary; also certify all smaller dimensions. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-27. The source thread verifies the property through dimension four using classifications of integral torsion conjugacy classes. The dated search found no general theorem or higher-dimensional counterexample.

On 2026-07-27 both MathOverflow answers and all their comments were checked. They reduce dimensions at most four to published lists and report ternary representatives in every listed class.
Tahara classifies the relevant low-dimensional finite subgroups, and Yang's 2015 Electronic Journal of Linear Algebra paper lists 45 torsion conjugacy classes in \(\operatorname{GL}_4(\mathbb Z)\). These checks set the first possible counterexample dimension at five.
Rational canonical form is insufficient because rational conjugacy can split into several integral conjugacy classes. A search must enumerate integral lattices or integral conjugacy classes.
A matrix outside the ternary alphabet is not itself a counterexample. One must certify that its entire \(\operatorname{GL}_n(\mathbb Z)\)-conjugacy class contains no ternary matrix.
Trap: allowing conjugation by \(\operatorname{GL}_n(\mathbb Q)\), changing the lattice, or using a larger bounded entry set answers a weaker question.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. Permutation matrices and signed permutation matrices already have entries in \(\{-1,0,1\}\).

Recorded example 2. The checked classifications supply the property for every torsion class in dimensions at most four.

2See also

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Plain text

“Ternary representatives of finite-order integral matrices.” TheoremDB. P2858. Problem statement; statement text SHA-256 e5f505439e90f75ba71c61aa3c64e1a107aa0f1a0d98325b7bf7e2f28a1c09d7. https://theoremdb.org/statement/?ref=P2858

BibTeX

@misc{theoremdb-problem-e5f505439e90f75ba71c61aa3c64e1a107aa0f1a0d98325b7bf7e2f28a1c09d7,
  title = {{Ternary representatives of finite-order integral matrices}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 e5f505439e90f75ba71c61aa3c64e1a107aa0f1a0d98325b7bf7e2f28a1c09d7},
  url = {https://theoremdb.org/statement/?ref=P2858}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1References

Packet source. MathOverflow question 346842, “Ternary representatives of finite-order integral matrices,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Original CC0 least-counterexample formulation written after reading both answers and checking the cited classifications through dimension four.Also cited at Full question, answers, and visible comments concerning Ternary representatives of finite-order integral matrices; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Ternary representatives of finite-order integral matrices, the reviewed source scope is Full question, answers, and visible comments concerning Ternary representatives of finite-order integral matrices; checked 2026-08-01.. The packet makes no inference beyond that cited scope.Source named by the research packet.
Qingjie Yang, “Conjugacy classes of torsion in GL_n(Z),” Electronic Journal of Linear Algebra 30 (2015), 478-493. DOI 10.13001/1081-3810.3110. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗website · reference source · checked 2026-08-01Source use: original summary.Reused material: abstract and complete list of 45 torsion conjugacy classes in GL_4(Z), pp. 478-493.Reuse basis: fair use reviewed · rights holder: Qingjie Yang and the Electronic Journal of Linear Algebra · checked 2026-08-01 by Philip Weiss, TheoremDB staff.Required attribution: Qingjie Yang, “Conjugacy classes of torsion in GL_n(Z),” Electronic Journal of Linear Algebra 30 (2015), 478-493. DOI 10.13001/1081-3810.3110.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and complete list of 45 torsion conjugacy classes in GL_4(Z), pp. 478-493.Source used to assess the problem's recorded status.For Ternary representatives of finite-order integral matrices, this source provides the exhaustive dimension-four classification used to check ternary representatives through d=4.
MathOverflow question 383058, “Ternary representatives of finite-order integral matrices,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at question asking whether A^k is integrally conjugate to A when gcd(k, ord(A))=1.Source used to assess the problem's recorded status.For Ternary representatives of finite-order integral matrices, this source was excluded as status evidence because it asks a different conjugacy predicate and does not establish ternary representatives; it is retained to document source-review history.

Original CC0 textbook restatement.

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[#P2858] Ternary representatives of finite-order integral matrices

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