Algorithms for computing with nilpotent matrix groups over infinite domains

A. S. Detinko; D. L. Flannery

arxiv: 1907.06045 · v1 · pith:RQK2LRCMnew · submitted 2019-07-13 · 🧮 math.GR

Algorithms for computing with nilpotent matrix groups over infinite domains

A. S. Detinko , D. L. Flannery This is my paper

Pith reviewed 2026-05-24 22:07 UTC · model grok-4.3

classification 🧮 math.GR

keywords nilpotent matrix groupsinfinite fieldsnilpotency testingcomputational group theorymatrix group algorithmsstructural propertieslinear groups

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The pith

A practical algorithm tests nilpotency of matrix groups over infinite fields.

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

The paper develops methods for computing with matrix groups over infinite domains. It applies these methods to design a practical algorithm that determines whether a matrix group over an infinite field is nilpotent. The same framework supplies further algorithms to resolve structural questions about any nilpotent matrix group that passes the test. The procedures are stated to work when the groups are presented by explicit finite generators whose entries support effective arithmetic.

Core claim

The central claim is that practical algorithms exist for testing nilpotency and determining structural properties of nilpotent matrix groups defined over infinite fields and other domains.

What carries the argument

The nilpotency-testing algorithm for matrix groups over an infinite field, built on effective computation with finite generating sets.

If this is right

Nilpotency becomes decidable for matrix groups over fields such as the rationals when generators are supplied explicitly.
Once nilpotency is confirmed, further structural features of the group can be computed algorithmically.
Matrix groups over infinite domains that satisfy the nilpotency condition become amenable to concrete computation.

Where Pith is reading between the lines

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

The same computational methods may extend to deciding other properties such as solvability for the same class of groups.
The work opens the possibility of systematic enumeration or classification of nilpotent linear groups over number fields.
Adaptations could handle matrix groups defined over rings or local fields rather than global fields.

Load-bearing premise

The input groups are given by finite sets of explicit matrix generators whose entries allow the decision procedure to terminate.

What would settle it

A finitely generated matrix group over the rational numbers on which the procedure either fails to halt or returns the wrong answer about nilpotency.

read the original abstract

We develop methods for computing with matrix groups defined over a range of infinite domains, and apply those methods to the design of algorithms for nilpotent groups. In particular, we provide a practical algorithm to test nilpotency of matrix groups over an infinite field. We also provide algorithms that answer a number of structural questions for a given nilpotent matrix group. The algorithms have been implemented in GAP and MAGMA.

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

The paper gives a working algorithm to test nilpotency for matrix groups over infinite fields, with implementations in GAP and MAGMA.

read the letter

The main takeaway is that the authors supply a practical algorithm for deciding nilpotency of matrix groups given by generators over an infinite field, plus some related structural routines, all implemented and available in two major systems. This extends earlier work that was mostly restricted to finite fields. The implementations are the clearest sign that the methods terminate and run on actual examples. They rely on standard effective presentations—finite generating sets with entries in a computable ring—which is the usual setup in this corner of computational group theory, so the restriction is not a surprise. The abstract and description show no circular reasoning or hidden non-termination; the constructions are built from group-theoretic properties rather than fitted parameters. A small soft spot is that full verification of every edge case would require the detailed proofs and complexity analysis in the body, which are not visible here, but the software provides an independent check. This work is aimed at people who already compute with linear groups in algebra packages and need to handle infinite domains such as the rationals. It is narrow but concrete, so a reading group focused on computational algebra might find it worth discussing. It deserves peer review because the algorithmic claim is specific, the implementations give something to test, and the extension to infinite fields fills a documented gap.

Referee Report

0 major / 1 minor

Summary. The paper develops methods for computing with matrix groups over infinite domains and applies them to nilpotent groups. It provides a practical algorithm to test nilpotency of matrix groups over an infinite field, plus algorithms answering structural questions for a given nilpotent matrix group. All algorithms are implemented in GAP and MAGMA.

Significance. If the algorithms terminate and are correct as claimed, the work supplies effective computational tools for matrix groups over infinite fields, a setting where standard finite-field methods do not apply. The dual implementations in GAP and MAGMA constitute reproducible, practical evidence of utility and allow independent verification.

minor comments (1)

The abstract and introduction would benefit from a brief explicit statement of the precise input format assumed for the matrix generators (e.g., entries in a computable subring) to make the termination claim fully transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper develops constructive algorithms for nilpotency testing and structural questions on matrix groups over infinite domains, grounded in explicit generators and effective computability over computable subrings, with implementations in GAP/MAGMA. No equations, fitted parameters, or derivations reduce by construction to inputs; the central claims consist of algorithmic procedures whose termination and correctness rest on standard group theory rather than self-referential definitions or self-citation chains. The work is self-contained against external benchmarks of effective group presentations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard group-theoretic definitions of nilpotency and matrix multiplication with no new free parameters, axioms beyond ordinary mathematics, or invented entities.

axioms (1)

standard math Standard definitions and properties of nilpotent groups and matrix multiplication over fields.
Algorithms presuppose the usual algebraic axioms for groups and fields.

pith-pipeline@v0.9.0 · 5586 in / 1073 out tokens · 19975 ms · 2026-05-24T22:07:51.202050+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

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B. Assmann, Polycyclic presentations for matrix groups, Diplomarbeit, Technische Universit¨ at Braunschweig, 2003

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Assmann and B

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Symbolic Comput

, Computing polycyclic presentations for polycyclic ration al matrix groups, J. Symbolic Comput. 40 (2005), no. 6, 1269–1284

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Babai, R

L. Babai, R. Beals, J. Cai, G. Ivanyos, and E. M. Luks, Multiplicative equations over commuting matrices , Proceed- ings of the Seventh Annual ACM-SIAM Symposium on Discrete Al gorithms (Atlanta, GA, 1996), ACM, New Y ork (1996), pp. 498–507

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Babai, R

L. Babai, R. Beals, and D. N. Rockmore, Deciding ﬁniteness of matrix groups in deterministic polynomial time, Proc. of International Symposium on Symbolic and Algebraic Compu tation ISSAC ’93 (ACM Press), 1993, pp. 117–126

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Beals, Algorithms for matrix groups and the Tits alternative , 36th IEEE Symposium on the Foundations of Computer Science (Milwaukee, WI, 1995), J

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Bosma, J

W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language , J. Symbolic Comput. 24 (1997), no. 3-4, 235–265

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A. S. Detinko, B. Eick, and D. L. Flannery, Nilmat—Computing with nilpotent matrix groups , a refereed GAP 4 package

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A. S. Detinko and D. L. Flannery, Classiﬁcation of nilpotent primitive linear groups over ﬁn ite ﬁelds, Glasg. Math. J. 46 (2004), no. 3, 585–594

work page 2004
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, Locally nilpotent linear groups, Irish Math. Soc. Bull. (2005), no. 56, 37–51

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, Computing in nilpotent matrix groups , LMS J. Comput. Math. 9 (2006), 104–134 (electronic)

work page 2006
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Locally nilpotent linear groups

, Corrigendum to “Locally nilpotent linear groups” [Irish Ma th. Soc. Bull. No. 56 (2005), 37–51] , Irish Math. Soc. Bull. (2006), no. 57, 103

work page 2005
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Symbolic Comput

, Algorithms for computing with nilpotent matrix groups over inﬁnite domains , J. Symbolic Comput. 43 (2008), no. 1, 8–26

work page 2008
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Math., 2018

, Linear groups and computation , Expo. Math., 2018

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B. Eick, Computational group theory, Jahresbericht der DMV 107, Heft 3 (2005), 155–170

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B. Eick, W. Nickel, and M. Horn, Polycyclic—Computation with polycyclic groups, a refereed GAP 4 package

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The GAP group, GAP – Groups, Algorithms, Programming

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D. F. Holt, B. Eick, and E. A. O’Brien, Handbook of computational group theory , Chapman & Hall/CRC, Boca Raton, FL, 2005

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E. H. Lo, Finding intersections and normalizers in ﬁnitely generate d nilpotent groups , J. Symbolic Comput. 25 (1998), no. 1, 45–59

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E. H. Lo and G. Ostheimer, A practical algorithm for ﬁnding matrix representations fo r polycyclic groups, J. Sym- bolic Comput. 28 (1999), no. 3, 339–360

work page 1999
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E. M. Luks, Computing in solvable matrix groups , Proc. 33rd IEEE Symposium on the Foundations of Computer Science, 1992, pp. 111–120

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Ostheimer, Practical algorithms for polycyclic matrix groups , J

G. Ostheimer, Practical algorithms for polycyclic matrix groups , J. Symbolic Comput. 28 (1999), no. 3, 361–379

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R´ onyai, Computations in associative algebras , Groups and computation (New Brunswick, NJ, 1991), DIMACS Ser

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Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol

D. Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge Unive rsity Press, Cambridge, 1983

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C. C. Sims, Computation with ﬁnitely presented groups, Encyclopedia of Mathematics and Its Applications, vol. 48 , Cambridge University Press, Cambridge, 1994

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D. A. Suprunenko, Matrix groups, Transl. Math. Monogr., vol. 45, American Mathematical Soc iety, Providence, RI, 1976

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B. A. F. Wehrfritz, Inﬁnite linear groups , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 76 , Springer- V erlag, New Y ork-Heidelberg, 1973

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, Nilpotent subgroups of GL(n, Q) , Glasg. Math. J. 43 (2001), no. 3, 477–485

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[1] [1]

Assmann, Polycyclic presentations for matrix groups, Diplomarbeit, Technische Universit¨ at Braunschweig, 2003

B. Assmann, Polycyclic presentations for matrix groups, Diplomarbeit, Technische Universit¨ at Braunschweig, 2003

work page 2003

[2] [2]

Assmann and B

B. Assmann and B. Eick, Polenta—Polycyclic presentations for matrix groups, a refereed GAP 4 package

work page

[3] [3]

Symbolic Comput

, Computing polycyclic presentations for polycyclic ration al matrix groups, J. Symbolic Comput. 40 (2005), no. 6, 1269–1284

work page 2005

[4] [4]

, Testing polycyclicity of ﬁnitely generated rational matri x groups, Math. Comp. 76 (2007), no. 259, 1669– 1682

work page 2007

[5] [5]

Babai, R

L. Babai, R. Beals, J. Cai, G. Ivanyos, and E. M. Luks, Multiplicative equations over commuting matrices , Proceed- ings of the Seventh Annual ACM-SIAM Symposium on Discrete Al gorithms (Atlanta, GA, 1996), ACM, New Y ork (1996), pp. 498–507

work page 1996

[6] [6]

Babai, R

L. Babai, R. Beals, and D. N. Rockmore, Deciding ﬁniteness of matrix groups in deterministic polynomial time, Proc. of International Symposium on Symbolic and Algebraic Compu tation ISSAC ’93 (ACM Press), 1993, pp. 117–126

work page 1993

[7] [7]

Beals, Algorithms for matrix groups and the Tits alternative , 36th IEEE Symposium on the Foundations of Computer Science (Milwaukee, WI, 1995), J

R. Beals, Algorithms for matrix groups and the Tits alternative , 36th IEEE Symposium on the Foundations of Computer Science (Milwaukee, WI, 1995), J. Comput. System S ci. 58 (1999), no. 2, 260–279

work page 1995

[8] [8]

Bialostocki, The nilpotency class of the p -Sylow subgroups of GL(n, q) where (p, q) = 1 , Canad

A. Bialostocki, The nilpotency class of the p -Sylow subgroups of GL(n, q) where (p, q) = 1 , Canad. Math. Bull. 29 (1986), no. 2, 218–223

work page 1986

[9] [9]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language , J. Symbolic Comput. 24 (1997), no. 3-4, 235–265

work page 1997

[10] [10]

P . M. Cohn, Algebra. Vol. 2, second ed., John Wiley & Sons Ltd., Chichester, 1989

work page 1989

[11] [11]

A. S. Detinko, B. Eick, and D. L. Flannery, Nilmat—Computing with nilpotent matrix groups , a refereed GAP 4 package

work page

[12] [12]

A. S. Detinko and D. L. Flannery, Classiﬁcation of nilpotent primitive linear groups over ﬁn ite ﬁelds, Glasg. Math. J. 46 (2004), no. 3, 585–594

work page 2004

[13] [13]

, Locally nilpotent linear groups, Irish Math. Soc. Bull. (2005), no. 56, 37–51

work page 2005

[14] [14]

, Computing in nilpotent matrix groups , LMS J. Comput. Math. 9 (2006), 104–134 (electronic)

work page 2006

[15] [15]

Locally nilpotent linear groups

, Corrigendum to “Locally nilpotent linear groups” [Irish Ma th. Soc. Bull. No. 56 (2005), 37–51] , Irish Math. Soc. Bull. (2006), no. 57, 103

work page 2005

[16] [16]

Symbolic Comput

, Algorithms for computing with nilpotent matrix groups over inﬁnite domains , J. Symbolic Comput. 43 (2008), no. 1, 8–26

work page 2008

[17] [17]

Math., 2018

, Linear groups and computation , Expo. Math., 2018

work page 2018

[18] [18]

J. D. Dixon, The structure of linear groups , V an Nostrand Reinhold, London, 1971

work page 1971

[19] [19]

, The orbit-stabilizer problem for linear groups , Canad. J. Math. 37 (1985), no. 2, 238–259

work page 1985

[20] [20]

Eick, Computational group theory, Jahresbericht der DMV 107, Heft 3 (2005), 155–170

B. Eick, Computational group theory, Jahresbericht der DMV 107, Heft 3 (2005), 155–170

work page 2005

[21] [21]

B. Eick, W. Nickel, and M. Horn, Polycyclic—Computation with polycyclic groups, a refereed GAP 4 package

work page

[22] [22]

The GAP group, GAP – Groups, Algorithms, Programming

work page

[23] [23]

D. F. Holt, B. Eick, and E. A. O’Brien, Handbook of computational group theory , Chapman & Hall/CRC, Boca Raton, FL, 2005

work page 2005

[24] [24]

E. H. Lo, Finding intersections and normalizers in ﬁnitely generate d nilpotent groups , J. Symbolic Comput. 25 (1998), no. 1, 45–59

work page 1998

[25] [25]

E. H. Lo and G. Ostheimer, A practical algorithm for ﬁnding matrix representations fo r polycyclic groups, J. Sym- bolic Comput. 28 (1999), no. 3, 339–360

work page 1999

[26] [26]

E. M. Luks, Computing in solvable matrix groups , Proc. 33rd IEEE Symposium on the Foundations of Computer Science, 1992, pp. 111–120

work page 1992

[27] [27]

Ostheimer, Practical algorithms for polycyclic matrix groups , J

G. Ostheimer, Practical algorithms for polycyclic matrix groups , J. Symbolic Comput. 28 (1999), no. 3, 361–379

work page 1999

[28] [28]

R´ onyai, Computations in associative algebras , Groups and computation (New Brunswick, NJ, 1991), DIMACS Ser

L. R´ onyai, Computations in associative algebras , Groups and computation (New Brunswick, NJ, 1991), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 11, Amer. Ma th. Soc., Providence, RI, 1993, pp. 221–243

work page 1991

[29] [29]

Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol

D. Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge Unive rsity Press, Cambridge, 1983

work page 1983

[30] [30]

C. C. Sims, Computation with ﬁnitely presented groups, Encyclopedia of Mathematics and Its Applications, vol. 48 , Cambridge University Press, Cambridge, 1994

work page 1994

[31] [31]

D. A. Suprunenko, Matrix groups, Transl. Math. Monogr., vol. 45, American Mathematical Soc iety, Providence, RI, 1976

work page 1976

[32] [32]

B. A. F. Wehrfritz, Inﬁnite linear groups , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 76 , Springer- V erlag, New Y ork-Heidelberg, 1973

work page 1973

[33] [33]

, Nilpotent subgroups of GL(n, Q) , Glasg. Math. J. 43 (2001), no. 3, 477–485

work page 2001