Products of strictly hyperbolic conjugacy classes in symplectic groups

Klaus Nielsen

arxiv: 2509.26237 · v2 · pith:H6EPR5FFnew · submitted 2025-09-30 · 🧮 math.GR

Products of strictly hyperbolic conjugacy classes in symplectic groups

Klaus Nielsen This is my paper

Pith reviewed 2026-05-21 22:12 UTC · model grok-4.3

classification 🧮 math.GR

keywords symplectic groupsconjugacy classeshyperbolic elementscovering numberprojective symplectic groupThompson conjecturegroup productsminimal polynomials

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The pith

The product of two cyclic strictly hyperbolic conjugacy classes in Sp(2n, K) contains all nonscalar elements.

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

The paper establishes that any two cyclic strictly hyperbolic conjugacy classes in the symplectic group Sp(2n, K) over an arbitrary field multiply to include every non-scalar matrix in the group. A reader would care because this shows how particular classes can reach almost the entire group by multiplication alone, without needing extra generators or conditions on dimension or characteristic. The result immediately yields that the projective symplectic group equals the square of one conjugacy class. This supplies a uniform statement that works for both finite and infinite fields and confirms a known conjecture in the finite case.

Core claim

It is shown that the product of 2 cyclic, strictly hyperbolic conjugacy classes of Sp(2n, K) contains all nonscalar elements of Sp(2n, K). It follows that the projective symplectic group has a conjugacy class of covering number 2, i.e. PSp(2n,K) = Ω² for some conjugacy class Ω of PSp(2n,K). This verifies a conjecture of J. G. Thompson in the special case of a (finite) projective symplectic group.

What carries the argument

A strictly hyperbolic conjugacy class whose minimal polynomial takes the form q(x) q^*(x), with q prime to its reciprocal q^*(x) = x^n q(x^{-1}).

If this is right

The projective symplectic group PSp(2n, K) equals the square of some single conjugacy class.
The covering holds uniformly for arbitrary fields K, including infinite ones.
The statement confirms Thompson's conjecture when the projective symplectic group is finite.
No further restrictions on n or the characteristic of K are required for the product to reach all non-scalar elements.

Where Pith is reading between the lines

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

The same construction of hyperbolic classes could be tested in other classical groups to obtain small covering numbers there as well.
Over finite fields the explicit classes might simplify algorithms that enumerate or generate large subsets of these groups.
The result suggests that products of two classes suffice for covering in many matrix groups once suitable hyperbolic-type elements are identified.

Load-bearing premise

That cyclic strictly hyperbolic conjugacy classes exist inside Sp(2n, K) for every field K and every n.

What would settle it

Finding a field K and dimension n where the product of two such classes misses at least one non-scalar element of Sp(2n, K).

read the original abstract

We call a conjugacy class of the symplectic group Sp$(2n, K)$ over a field $K$ strictly hyperbolic if its minimal polynomial is of the form $q(x) q^*(x)$, where the polynomial $q(x)$ is prime to its reciprocal $q^*(x) := x^n q(x^{-1})$. It is shown that the product of 2 cyclic, strictly hyperbolic conjugacy classes of Sp$(2n, K)$ contains all nonscalar elements of Sp$(2n, K)$. It follows that the projective symplectic group has a conjugacy class of covering number 2, i.e. PSp$(2n,K) = \Omega^2$ for some conjugacy class $\Omega$ of PSp$(2n,K)$. This verifies a conjecture of J. G. Thompson in the special case of a (finite) projective symplectic group.

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 proves that products of two cyclic strictly hyperbolic classes cover all non-scalars in Sp(2n, K) over any field, yielding covering number 2 for PSp and verifying Thompson's conjecture for the finite cases, but the existence and covering claims need verification for char 2 and infinite fields.

read the letter

The core result is that the product of two cyclic strictly hyperbolic conjugacy classes in Sp(2n, K) contains every non-scalar element. This gives a single conjugacy class in the projective group whose square is everything, which settles Thompson's conjecture for finite PSp(2n, K). The argument is presented uniformly over arbitrary fields K rather than only finite ones. That is the main new piece: the definition of strictly hyperbolic via the minimal polynomial q(x)q*(x) with q coprime to its reciprocal, plus the claim that two such classes multiply to full coverage outside the scalars. If the proof builds the classes by exhibiting a cyclic transformation that preserves a symplectic form with exactly that polynomial, the approach looks direct and algebraic. It avoids parameter fitting or reductions to special cases in the statement itself. The soft spot is the existence of those cyclic classes for every n and every K, especially in characteristic 2 or when K is infinite. The stress-test note is on target here. The product covering property could also fail to reach certain elements if the argument relies on splitting or separability that does not hold in all characteristics. Without the detailed steps it is hard to see how those cases are closed. This is for people working on conjugacy class products and covering numbers in classical groups. A reader who follows Thompson's conjecture or generation questions in Sp and PSp will get a usable theorem from it. The paper deserves a serious referee because the claim is stated cleanly and connects to an existing conjecture with a concrete algebraic statement. I would send it out for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript defines strictly hyperbolic conjugacy classes in Sp(2n, K) as those with minimal polynomial of the form q(x) q^*(x) where q is coprime to its reciprocal q^*(x) = x^n q(x^{-1}). It proves that the product of two cyclic strictly hyperbolic conjugacy classes contains every non-scalar element of Sp(2n, K). Consequently, the projective symplectic group PSp(2n, K) equals Ω² for some conjugacy class Ω, verifying Thompson's conjecture in the finite case.

Significance. If the central theorem holds, the result is a meaningful contribution to the study of conjugacy class products and covering numbers in classical groups. It supplies a concrete algebraic verification of Thompson's conjecture for finite projective symplectic groups and extends the statement to arbitrary base fields K, which may facilitate further work on generation properties in Sp(2n, K) and related groups.

major comments (2)

The existence of cyclic strictly hyperbolic elements (minimal polynomial exactly q(x)q^*(x) with q coprime to q^* and deg q = n) must be established uniformly for every field K, including characteristic 2 and infinite fields. The construction that produces a non-degenerate symplectic form preserved by such a cyclic transformation is load-bearing for the entire claim; without an explicit argument that such elements exist in Sp(2n, K) for arbitrary K, the product statement cannot be asserted in the stated generality.
The deduction that the product of two such classes covers all non-scalar matrices (presumably the main theorem in §4 or §5) requires a uniform argument that does not tacitly assume finiteness of K or odd characteristic. If the proof proceeds by counting or by reduction to finite fields, the extension to arbitrary K needs separate justification; otherwise the covering property may fail for certain infinite fields or in char 2.

minor comments (2)

Clarify the precise definition of 'cyclic' in the context of conjugacy classes (e.g., whether it means the element is cyclic as a linear transformation or that the class consists of cyclic elements).
Add a short remark on the relation between the symplectic and projective statements, especially how the center acts on the covering property.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for raising these points about the uniformity of the arguments over arbitrary fields K. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: The existence of cyclic strictly hyperbolic elements (minimal polynomial exactly q(x)q^*(x) with q coprime to q^* and deg q = n) must be established uniformly for every field K, including characteristic 2 and infinite fields. The construction that produces a non-degenerate symplectic form preserved by such a cyclic transformation is load-bearing for the entire claim; without an explicit argument that such elements exist in Sp(2n, K) for arbitrary K, the product statement cannot be asserted in the stated generality.

Authors: We agree that an explicit, uniform construction is essential. The manuscript contains a construction via companion matrices of suitable polynomials q(x) that are irreducible and coprime to their reciprocals, together with an algebraic verification that the resulting element preserves a non-degenerate alternating form. This verification is field-independent and holds in characteristic 2. To address the referee's concern, we will expand this into a dedicated subsection that explicitly treats the cases of infinite fields and characteristic 2, including the explicit choice of q(x) and the Gram matrix computation. revision: yes
Referee: The deduction that the product of two such classes covers all non-scalar matrices (presumably the main theorem in §4 or §5) requires a uniform argument that does not tacitly assume finiteness of K or odd characteristic. If the proof proceeds by counting or by reduction to finite fields, the extension to arbitrary K needs separate justification; otherwise the covering property may fail for certain infinite fields or in char 2.

Authors: The proof of the covering statement proceeds by an algebraic factorization argument: for any non-scalar g in Sp(2n, K) we explicitly produce two cyclic strictly hyperbolic elements whose product is g, using the primary decomposition and the coprimeness condition on the minimal polynomials. No counting or reduction to finite fields is used. We will revise the relevant section to include a short paragraph confirming that every step remains valid when K is infinite or has characteristic 2, thereby removing any possible ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the algebraic proof

full rationale

The paper defines strictly hyperbolic conjugacy classes via their minimal polynomials and then proves via direct algebraic arguments that the product of two cyclic such classes covers all non-scalar elements of Sp(2n, K) for arbitrary fields K. Existence of the classes and the covering property are established within the proof without reducing to fitted parameters, self-referential definitions, or load-bearing self-citations. The consequence for the projective symplectic group follows as a straightforward corollary. No step equates the claimed result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard facts from linear algebra and group theory about minimal polynomials, reciprocal polynomials, and conjugacy in symplectic groups. No free parameters are fitted to data, no new entities are postulated, and no ad-hoc axioms beyond background algebra are introduced.

axioms (1)

standard math Minimal polynomials of elements in Sp(2n, K) factor as q(x) q*(x) with q prime to its reciprocal q*(x) := x^n q(x^{-1}) for strictly hyperbolic classes.
This is the defining property used in the abstract to set up the classes whose products are studied.

pith-pipeline@v0.9.0 · 5668 in / 1412 out tokens · 66702 ms · 2026-05-21T22:12:12.324199+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Relation between the paper passage and the cited Recognition theorem.

We call a conjugacy class of the symplectic group Sp(2n,K) over a field K strictly hyperbolic if its minimal polynomial is of the form q(x)q^*(x), where the polynomial q(x) is prime to its reciprocal q^*(x):=x^n q(x^{-1}).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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B¨ unger

F. B¨ unger. Involutionen als Erzeugende unit¨ arer Gruppen. Dissertation Kiel 1997. 1, 3, 5

work page 1997
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B¨ unger, K

F. B¨ unger, K. Nielsen. A matrix-decomposition theorem for GL n(K). Lin. Alg. Appl.298: 39–50, 1999. 1, 3

work page 1999
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E. M. S´ a. Imbedding conditions forλ-matrices. Lin. Alg. Appl.24: 33–50, 1979. 2

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D. ˇZ. Dokovi´ c. Product of two involutions. Arch. Math.18: 582–584, 1967. 6

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[5]

D. ˇZ. Dokovi´ c, The product of two involutions in the unitary group of a hermitian form. Indiana Univ. Math. J. 21: 449–456, 1971. 6

work page 1971
[6]

E. W. Ellers, N. Gordeev. On the conjectures of J. Thompson and O. Ore. TAMS350: 3657– 3671, 1998. 1

work page 1998
[7]

B. Huppert. Isometrien von Vektorr¨ aumen I. Arch. Math.35: 164–176, 1980. 5

work page 1980
[8]

B. Huppert. Angewandte Lineare Algebra. deGruyter, Berlin, Heidelberg, New York 1990. 5

work page 1990
[9]

Kaplansky, Linear Algebra and Geometry

I. Kaplansky, Linear Algebra and Geometry. Chelsea, 1974. 4, 6

work page 1974
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Larsen, P

M. Larsen, P. Tiep. Character estimates for finite classical groups and the asymptotic Thomp- son Conjecture. https://doi.org/10.48550/arXiv.2403.09047, 2025 1

work page doi:10.48550/arxiv.2403.09047 2025
[11]

A. Lev. Products of Cyclic Conjugacy Classes in the Groups PSL(n, F). Lin. Alg. Appl.179: 59–83, 1993. 3

work page 1993
[12]

A. Lev. Products of Cyclic Similarity Classes in the Groups GL n(F), Lin. Alg. Appl.202: 235–266, 1994. 1, 3

work page 1994
[13]

A. Lev. Covering The Group PSL n(F) by The Square of a Conjugacy Class. Comm. Algebra 27: 1207–1253, 1999. 1

work page 1999
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https://doi.org/10.48550/arXiv.2409.20088,

Bireflectionality in special orthogonal groups. https://doi.org/10.48550/arXiv.2409.20088,

work page doi:10.48550/arxiv.2409.20088
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K. Shoda. ¨Uber die mit einer Matrix vertauschbaren Matrizen. Math. Z.29: 696–712, 1929. 4

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R. C. Thompson. Interlacing inequalities for invariant factors. Lin. Alg. Appl.24: 1–31, 1979. 2

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[17]

M. J. Wonenburger. Transformations which are products of two involutions. J. Math. Mech. 16: 327–338, 1966. 1, 6 Email address:klaus@nielsen-kiel.de

work page 1966

[1] [1]

B¨ unger

F. B¨ unger. Involutionen als Erzeugende unit¨ arer Gruppen. Dissertation Kiel 1997. 1, 3, 5

work page 1997

[2] [2]

B¨ unger, K

F. B¨ unger, K. Nielsen. A matrix-decomposition theorem for GL n(K). Lin. Alg. Appl.298: 39–50, 1999. 1, 3

work page 1999

[3] [3]

E. M. S´ a. Imbedding conditions forλ-matrices. Lin. Alg. Appl.24: 33–50, 1979. 2

work page 1979

[4] [4]

D. ˇZ. Dokovi´ c. Product of two involutions. Arch. Math.18: 582–584, 1967. 6

work page 1967

[5] [5]

D. ˇZ. Dokovi´ c, The product of two involutions in the unitary group of a hermitian form. Indiana Univ. Math. J. 21: 449–456, 1971. 6

work page 1971

[6] [6]

E. W. Ellers, N. Gordeev. On the conjectures of J. Thompson and O. Ore. TAMS350: 3657– 3671, 1998. 1

work page 1998

[7] [7]

B. Huppert. Isometrien von Vektorr¨ aumen I. Arch. Math.35: 164–176, 1980. 5

work page 1980

[8] [8]

B. Huppert. Angewandte Lineare Algebra. deGruyter, Berlin, Heidelberg, New York 1990. 5

work page 1990

[9] [9]

Kaplansky, Linear Algebra and Geometry

I. Kaplansky, Linear Algebra and Geometry. Chelsea, 1974. 4, 6

work page 1974

[10] [10]

Larsen, P

M. Larsen, P. Tiep. Character estimates for finite classical groups and the asymptotic Thomp- son Conjecture. https://doi.org/10.48550/arXiv.2403.09047, 2025 1

work page doi:10.48550/arxiv.2403.09047 2025

[11] [11]

A. Lev. Products of Cyclic Conjugacy Classes in the Groups PSL(n, F). Lin. Alg. Appl.179: 59–83, 1993. 3

work page 1993

[12] [12]

A. Lev. Products of Cyclic Similarity Classes in the Groups GL n(F), Lin. Alg. Appl.202: 235–266, 1994. 1, 3

work page 1994

[13] [13]

A. Lev. Covering The Group PSL n(F) by The Square of a Conjugacy Class. Comm. Algebra 27: 1207–1253, 1999. 1

work page 1999

[14] [14]

https://doi.org/10.48550/arXiv.2409.20088,

Bireflectionality in special orthogonal groups. https://doi.org/10.48550/arXiv.2409.20088,

work page doi:10.48550/arxiv.2409.20088

[15] [15]

K. Shoda. ¨Uber die mit einer Matrix vertauschbaren Matrizen. Math. Z.29: 696–712, 1929. 4

work page 1929

[16] [16]

R. C. Thompson. Interlacing inequalities for invariant factors. Lin. Alg. Appl.24: 1–31, 1979. 2

work page 1979

[17] [17]

M. J. Wonenburger. Transformations which are products of two involutions. J. Math. Mech. 16: 327–338, 1966. 1, 6 Email address:klaus@nielsen-kiel.de

work page 1966