# P2858: Ternary representatives of finite-order integral matrices

- ID: `P2858`
- Reference: `finite-order-integer-matrix-ternary-conjugate`
- Page: https://theoremdb.org/statements/P2858
- Record maturity: Reviewed problem with recorded work

## 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.

### Definitions

- **Definition.** Finite order means that \(A^r=I_n\) for some positive integer \(r\).
- **Definition.** Integral conjugacy means replacing \(A\) by \(PAP^{-1}\) for some \(P\in\operatorname{GL}_n(\mathbb Z)\).
- **Definition.** A ternary matrix is an integer matrix all of whose entries belong to \(\{-1,0,1\}\).

### What 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.

## Status

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](#reference-2) [1](#reference-1) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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

- 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.

- Recorded example: Permutation matrices and signed permutation matrices already have entries in \(\{-1,0,1\}\).
- Recorded example: The checked classifications supply the property for every torsion class in dimensions at most four.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `finite-order-integer-matrix-ternary-conjugate`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 346842, “Ternary representatives of finite-order integral matrices,” checked 2026-08-01. Original CC0 least-counterexample formulation written after reading both answers and checking the cited classifications through dimension four. https://mathoverflow.net/questions/346842/is-every-finite-order-unimodular-matrix-conjugate-to-a-0-1-1-matrix
   - 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.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>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. abstract and complete list of 45 torsion conjugacy classes in GL_4(Z), pp. 478-493 https://journals.uwyo.edu/index.php/ela/article/view/1561/1561
   - website; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
   - 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.
3. <a id="reference-3"></a>MathOverflow question 383058, “Ternary representatives of finite-order integral matrices,” checked 2026-08-01. question asking whether A^k is integrally conjugate to A when gcd(k, ord(A))=1 https://mathoverflow.net/questions/383058/finite-order-elements-of-mathrmgl-dmathbbz-that-are-conjugate-to-powers
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.