On the proportion of derangements in affine classical groups

Jessica Anzanello

arxiv: 2508.07093 · v2 · submitted 2025-08-09 · 🧮 math.CO · math.GR

On the proportion of derangements in affine classical groups

Jessica Anzanello This is my paper

Pith reviewed 2026-05-18 23:40 UTC · model grok-4.3

classification 🧮 math.CO math.GR

keywords derangementsaffine classical groupsfinite fieldsproportionspartitionsq-polynomialsunitary groupssymplectic groups

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The pith

Exact formulas are derived for the proportions of derangements and p-power derangements in affine classical groups over finite fields.

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

The paper establishes closed-form expressions for the share of derangements among elements of the affine unitary, symplectic and orthogonal groups. Derangements are the elements that fix no vectors in the natural module. These proportions matter for understanding how often a random element from such a group acts without fixed points, a quantity that arises in generation problems and in counting orbit sizes. For the unitary groups the formulas rest on a new generating function that enumerates a restricted family of integer partitions; for the remaining families the work reduces to checking three polynomial identities in the variable q.

Core claim

We derive exact formulas for the proportions of derangements and of derangements of p-power order in the affine classical groups AU_m(q), ASp_{2m}(q), AO_{2m+1}(q) and AO^±_{2m}(q). In the unitary case these formulas depend on a generating function for partitions of length m that satisfy either λ_1 = 1 or λ_{k-1} > λ_k = k for some index k. In the symplectic and orthogonal cases the derivations reduce to the verification of three q-polynomial identities that were conjectured earlier and later confirmed.

What carries the argument

The generating function for m-part partitions obeying λ_1=1 or λ_{k-1}>λ_k=k for some k (unitary case) together with the three q-polynomial identities (symplectic and orthogonal cases).

If this is right

The formulas permit exact computation of the probability that a random element of any of these groups fixes no vector.
Separate formulas for p-power-order derangements give the density of unipotent fixed-point-free elements.
The partition generating function can be studied on its own as a contribution to enumeration of restricted partitions.
The verified q-polynomial identities supply new relations among cycle indices or character values in the affine groups.

Where Pith is reading between the lines

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

The same proportions could be inserted into probabilistic generation algorithms to bound the chance that a random pair of elements generates the full affine group.
Asymptotic limits of the formulas as the field size q grows with m fixed would give the natural density of fixed-point-free elements in the infinite affine groups over algebraically closed fields.
The partition condition λ_{k-1} > λ_k = k may correspond to a concrete cycle-type restriction that could be rephrased in terms of the support of the translation part of an affine element.

Load-bearing premise

The three q-polynomial identities hold for the symplectic and orthogonal families.

What would settle it

Enumerate all elements of AU_2(3) or ASp_2(3) by computer, count the derangements directly, and check whether the resulting proportion equals the closed formula given for that small case.

Figures

Figures reproduced from arXiv: 2508.07093 by Jessica Anzanello.

**Figure 1.** Figure 1: Example: λ = (8, 7, 7, 4, 4, 3, 3, 1, 1) has Durfee square 4, π1(λ) = (4, 3, 3), π2(λ) = (4, 3, 3, 1, 1) and it satisfies λ3 > λ4 = 4. We now prove Theorem 1.2, assuming Theorem 1.3, which will be established in Section 3.3. Proof of Theorem 1.2. From Lemma 3.1 we have um(q) = (−1)m X |λ|=m 1 − q −2λ ′ 1 (−q) P i (λ ′ i ) 2 Q i (−1/q)mi(λ) , and from Eq. (11) and Lemma 3.3 we have um(q) = (−1)mH(−1/q), whe… view at source ↗

**Figure 2.** Figure 2: Graphical representation of Φ(a, b) • η4 fits in the strip u × ∞, • |η1| + |η2| = |η3| + |η4|. The existence of such a bijection is guaranteed by the Gaussian binomial identity, which states that u + v u q = (q)u+v (q)u(q)v , where the q-binomial coefficient u+v u q is the generating function for partitions whose Ferrers diagram fits inside a u × v rectangle, and 1 (q)u is the generating function for… view at source ↗

**Figure 3.** Figure 3: Now, for A(x), the choices at point (i) and (ii), correspond to selecting a partition µ with maximal part equal to ak. Indeed, for such a partition, let k be the size of its Durfee square, then a1, . . . , ak correspond to the part of the Ferrers diagram above the main diagonal, and b1, ..., bk correspond to the part below it (as in the first example of [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: Example of a symplectic signed partition: here, the + corresponds to the part of size 8 and the − corresponds to the parts of size 4 and 2 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: Example of an orthogonal signed partition: here the + corresponds to the part of size 3 and the − corresponds to the parts of size 5 and 1. Theorem 5.1 ([Ful99], Theorem 13). The data λ ± z−1 , λ± z+1, λϕ represent a conjugacy class of some orthogonal group if (i) |λz| = 0, (ii) λϕ = λϕ¯, (iii) P ϕ=z±1 |λ ± ϕ | + P ϕ̸=z±1 |λϕ| deg(ϕ) = m. In this case, these data represent the conjugacy class of exactly on… view at source ↗

read the original abstract

We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, where $p$ denotes the characteristic of the defining finite field. In the unitary case, the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $\lambda=(\lambda_1, \dots, \lambda_m)$ into $m$ parts, with $\lambda_1\ge \dots \ge \lambda_m$, such that either $\lambda_1=1$ or $\lambda_{k-1}>\lambda_k=k$ for some $k \in \{2, \dots,m\}$. In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three $q$-polynomial identities conjectured by the author and later proved by Fulman and Stanton.

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 delivers exact closed-form proportions for derangements and p-power derangements across the four families of affine classical groups, with a fresh generating function for restricted partitions in the unitary case.

read the letter

The main point is that this work gives explicit formulas for the proportion of derangements (and those of p-power order) in AU_m(q), ASp_{2m}(q), AO_{2m+1}(q), and AO^±_{2m}(q). Those formulas are the central output and appear to be new for the affine setting. The unitary case stands on its own with a generating function for partitions into m parts satisfying either λ1=1 or λ_{k-1}>λ_k=k for some k; that combinatorial statement is presented clearly and looks like an independent contribution worth noting on its own. The other three families reduce the counting problem to three q-polynomial identities that the author conjectured and Fulman-Stanton later proved, so the paper's role there is mainly to establish the correct mapping from group elements to those identities. That keeps the argument short and avoids duplicating the polynomial work. The reductions are described explicitly enough in the abstract and setup that a reader can check the logic without too much trouble. The main soft spot is that the symplectic and orthogonal formulas inherit any limitations in how the fixed-point-free counts translate into the left-hand sides of those identities. If the substitution or the accounting of p-power order elements has an off-by-one or a missed conjugacy class, the closed forms would need adjustment even though the identities themselves are settled. The paper does not appear to introduce circularity beyond citing those external proofs, and the unitary generating function is handled separately. The citation pattern to earlier derangement results in classical groups is standard and appropriate. This is a paper for people who need exact expressions rather than asymptotics in probabilistic group theory or finite groups of Lie type. A reader working on derangement probabilities or on q-series identities tied to groups of Lie type will get direct value from the formulas and the partition result. It has enough new content and formal grounding to deserve referee time rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The manuscript derives exact formulas for the proportions of derangements and of derangements of p-power order in the affine classical groups AU_m(q), ASp_{2m}(q), AO_{2m+1}(q) and AO^±_{2m}(q). In the unitary case the formulas rest on a new generating function for integer partitions into m parts satisfying either λ_1=1 or λ_{k-1}>λ_k=k for some k; in the symplectic and orthogonal cases the derivations reduce to verifying three q-polynomial identities conjectured by the author and subsequently proved by Fulman and Stanton.

Significance. If the reductions are accurate, the results supply closed-form expressions for these proportions, which are of interest in the study of random elements and fixed-point-free permutations in affine groups of Lie type. The partition generating function is of independent combinatorial interest and strengthens the unitary contribution.

major comments (1)

[Sections deriving the formulas for ASp_{2m}(q), AO_{2m+1}(q) and AO^±_{2m}(q)] The symplectic and orthogonal derivations map counts of fixed-point-free elements (and those of p-power order) in the affine groups to the left-hand sides of the three q-polynomial identities. The manuscript must explicitly confirm that this substitution introduces no algebraic or combinatorial mismatch (for example, in the treatment of fixed spaces or the precise generating-function replacement), since any such discrepancy would render the claimed closed forms invalid even though the external identities hold.

minor comments (2)

Add a short table or explicit low-dimensional check (e.g., m=2, small q) that directly compares the derived formula against brute-force enumeration of derangements in the affine group to illustrate the reduction.
Ensure the reference list contains the full bibliographic details for the Fulman–Stanton proofs of the q-polynomial identities.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our results and for their detailed major comment. We address the concern regarding the symplectic and orthogonal derivations below.

read point-by-point responses

Referee: [Sections deriving the formulas for ASp_{2m}(q), AO_{2m+1}(q) and AO^±_{2m}(q)] The symplectic and orthogonal derivations map counts of fixed-point-free elements (and those of p-power order) in the affine groups to the left-hand sides of the three q-polynomial identities. The manuscript must explicitly confirm that this substitution introduces no algebraic or combinatorial mismatch (for example, in the treatment of fixed spaces or the precise generating-function replacement), since any such discrepancy would render the claimed closed forms invalid even though the external identities hold.

Authors: We appreciate the referee's call for explicit confirmation to ensure the validity of the closed forms. In the original manuscript, the derivations in the relevant sections proceed by expressing the proportion of derangements as a certain sum or product that directly corresponds to the left-hand side of the q-polynomial identities via the cycle index or the generating function for the number of elements with given fixed space dimension. The substitution is justified because the affine action's fixed-point condition reduces precisely to the linear case adjusted for the translation, and the p-power order condition aligns with the nilpotency or Jordan block structures accounted for in the identities. To address the referee's point directly and enhance clarity, we will add a dedicated paragraph in the revised version explicitly stating that there is no mismatch in the treatment of fixed spaces, as the generating function replacement is one-to-one with the combinatorial objects counted in the affine group. revision: yes

Circularity Check

0 steps flagged

Derivations for affine classical groups rely on independent generating functions and external proofs of q-polynomial identities

full rationale

The paper derives exact formulas for derangement proportions in AU_m(q) via a new generating function for restricted partitions that is developed independently within the manuscript. For ASp_{2m}(q), AO_{2m+1}(q) and AO^±_{2m}(q), the proofs reduce the counting problems to three specific q-polynomial identities; these identities were conjectured by the present author but are cited as having been proved by the distinct authors Fulman and Stanton. Because the load-bearing steps invoke externally proved results rather than self-referential definitions, fitted parameters renamed as predictions, or ansatzes smuggled through overlapping citations, the derivation chain remains self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard facts from finite group theory and combinatorics without introducing fitted parameters or new entities.

axioms (1)

standard math Standard definitions and properties of affine classical groups and the notion of derangement in their natural action on affine space.
The derivations presuppose established facts about the groups AU_m(q), ASp_{2m}(q), AO_{2m+1}(q) and AO^±_{2m}(q).

pith-pipeline@v0.9.0 · 5697 in / 1350 out tokens · 83519 ms · 2026-05-18T23:40:43.381795+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The proofs of the formulas reduce to verifying three q-polynomial identities conjectured by the author and later proved by Fulman and Stanton.

What do these tags mean?

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

[BG16] Timothy C

Reprint of the 1976 original. [BG16] Timothy C. Burness and Michael Giudici. Classical groups, derangements and primes , volume 25 of Australian Mathematical Society Lecture Series . Cambridge University Press, Cambridge,

work page 1976
[2]

[BK22] Aubrey Blecher and Arnold Knopfmacher

doi: 10.1017/CBO9781139059060. [BK22] Aubrey Blecher and Arnold Knopfmacher. Fixed points and matching points in partitions. Ramanujan J., 58(1):23–41,

work page doi:10.1017/cbo9781139059060
[3]

[Bri06] John R

doi:10.1007/s11139-022-00551-x . [Bri06] John R. Britnell. Cyclic, separable and semisimple transformations in the finite conformal groups. J. Group Theory, 9(5):571–601,

work page doi:10.1007/s11139-022-00551-x
[4]

[dM80] Pierre R´ emond de Montmort.Essay d’analyse sur les jeux de hazard

doi:10.1515/JGT.2006.038. [dM80] Pierre R´ emond de Montmort.Essay d’analyse sur les jeux de hazard . Chelsea Publishing Co., New York, second edition,

work page doi:10.1515/jgt.2006.038 2006
[5]

Trans- lated from the Latin and with an introduction by John D. Blanton. doi:10.1007/978-1-4612-1001-6 . [FG03] Jason Fulman and Robert Guralnick. Derangements in simple and primitive groups. In Groups, com- binatorics & geometry (Durham,

work page doi:10.1007/978-1-4612-1001-6
[6]

[FG12] Jason Fulman and Robert Guralnick

doi:10.1142/9789812564481\_0006. [FG12] Jason Fulman and Robert Guralnick. Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements. Trans. Amer. Math. Soc., 364(6):3023–3070,

work page doi:10.1142/9789812564481
[7]

[FG17] Jason Fulman and Robert Guralnick

doi:10.1090/S0002-9947-2012-05427-4 . [FG17] Jason Fulman and Robert Guralnick. Derangements in subspace actions of finite classical groups. Trans. Amer. Math. Soc., 369(4):2521–2572,

work page doi:10.1090/s0002-9947-2012-05427-4 2012
[8]

[FG18] Jason Fulman and Robert Guralnick

doi:10.1090/tran/6721. [FG18] Jason Fulman and Robert Guralnick. Derangements in finite classical groups for actions related to ex- tension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture. Trans. Amer. Math. Soc., 370(7):4601–4622,

work page doi:10.1090/tran/6721
[9]

[Ful99] Jason Fulman

doi:10.1090/tran/7377. [Ful99] Jason Fulman. Cycle indices for the finite classical groups. J. Group Theory , 2(3):251–289,

work page doi:10.1090/tran/7377
[10]

[GT03] Robert M

doi: 10.1515/jgth.1999.017. [GT03] Robert M. Guralnick and Pham Huu Tiep. Finite simple unisingular groups of Lie type. J. Group Theory, 6(3):271–310,

work page doi:10.1515/jgth.1999.017 1999
[11]

[Hop24] Brian Hopkins

doi:10.1515/jgth.2003.020. [Hop24] Brian Hopkins. Refining Blecher and Knopfmacher’s integer partition fixed points.Enumer. Comb. Appl., 4(4):Paper No. S2R26, 6,

work page doi:10.1515/jgth.2003.020 2003
[12]

[HS24] Brian Hopkins and James A

doi:10.54550/eca2024v4s4r26. [HS24] Brian Hopkins and James A. Sellers. On Blecher and Knopfmacher’s fixed points for integer partitions. Discrete Math., 347(5):Paper No. 113938, 9,

work page doi:10.54550/eca2024v4s4r26
[13]

[NP98] Peter M

doi:10.1016/j.disc.2024.113938. [NP98] Peter M. Neumann and Cheryl E. Praeger. Derangements and eigenvalue-free elements in finite classical groups. J. London Math. Soc. (2) , 58(3):564–586,

work page doi:10.1016/j.disc.2024.113938 2024
[14]

[Pet] Fedor Petrov

doi:10.1112/S0024610798006772. [Pet] Fedor Petrov. A bijection between pairs of partitions. MathOverflow. URL: https://mathoverflow.net/ q/487338. [Ser03] Jean-Pierre Serre. On a theorem of Jordan. Bull. Amer. Math. Soc. (N.S.) , 40(4):429–440,

work page doi:10.1112/s0024610798006772
[15]

[Spi17] Pablo Spiga

doi: 10.1090/S0273-0979-03-00992-3 . [Spi17] Pablo Spiga. On the number of derangements and derangements of prime power order of the affine general linear groups. J. Algebraic Combin., 45(2):345–362,

work page doi:10.1090/s0273-0979-03-00992-3
[16]

[Ste68] Robert Steinberg

doi:10.1007/s10801-016-0709-3 . [Ste68] Robert Steinberg. Endomorphisms of linear algebraic groups, volume No. 80 of Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI,

work page doi:10.1007/s10801-016-0709-3

[1] [1]

[BG16] Timothy C

Reprint of the 1976 original. [BG16] Timothy C. Burness and Michael Giudici. Classical groups, derangements and primes , volume 25 of Australian Mathematical Society Lecture Series . Cambridge University Press, Cambridge,

work page 1976

[2] [2]

[BK22] Aubrey Blecher and Arnold Knopfmacher

doi: 10.1017/CBO9781139059060. [BK22] Aubrey Blecher and Arnold Knopfmacher. Fixed points and matching points in partitions. Ramanujan J., 58(1):23–41,

work page doi:10.1017/cbo9781139059060

[3] [3]

[Bri06] John R

doi:10.1007/s11139-022-00551-x . [Bri06] John R. Britnell. Cyclic, separable and semisimple transformations in the finite conformal groups. J. Group Theory, 9(5):571–601,

work page doi:10.1007/s11139-022-00551-x

[4] [4]

[dM80] Pierre R´ emond de Montmort.Essay d’analyse sur les jeux de hazard

doi:10.1515/JGT.2006.038. [dM80] Pierre R´ emond de Montmort.Essay d’analyse sur les jeux de hazard . Chelsea Publishing Co., New York, second edition,

work page doi:10.1515/jgt.2006.038 2006

[5] [5]

Trans- lated from the Latin and with an introduction by John D. Blanton. doi:10.1007/978-1-4612-1001-6 . [FG03] Jason Fulman and Robert Guralnick. Derangements in simple and primitive groups. In Groups, com- binatorics & geometry (Durham,

work page doi:10.1007/978-1-4612-1001-6

[6] [6]

[FG12] Jason Fulman and Robert Guralnick

doi:10.1142/9789812564481\_0006. [FG12] Jason Fulman and Robert Guralnick. Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements. Trans. Amer. Math. Soc., 364(6):3023–3070,

work page doi:10.1142/9789812564481

[7] [7]

[FG17] Jason Fulman and Robert Guralnick

doi:10.1090/S0002-9947-2012-05427-4 . [FG17] Jason Fulman and Robert Guralnick. Derangements in subspace actions of finite classical groups. Trans. Amer. Math. Soc., 369(4):2521–2572,

work page doi:10.1090/s0002-9947-2012-05427-4 2012

[8] [8]

[FG18] Jason Fulman and Robert Guralnick

doi:10.1090/tran/6721. [FG18] Jason Fulman and Robert Guralnick. Derangements in finite classical groups for actions related to ex- tension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture. Trans. Amer. Math. Soc., 370(7):4601–4622,

work page doi:10.1090/tran/6721

[9] [9]

[Ful99] Jason Fulman

doi:10.1090/tran/7377. [Ful99] Jason Fulman. Cycle indices for the finite classical groups. J. Group Theory , 2(3):251–289,

work page doi:10.1090/tran/7377

[10] [10]

[GT03] Robert M

doi: 10.1515/jgth.1999.017. [GT03] Robert M. Guralnick and Pham Huu Tiep. Finite simple unisingular groups of Lie type. J. Group Theory, 6(3):271–310,

work page doi:10.1515/jgth.1999.017 1999

[11] [11]

[Hop24] Brian Hopkins

doi:10.1515/jgth.2003.020. [Hop24] Brian Hopkins. Refining Blecher and Knopfmacher’s integer partition fixed points.Enumer. Comb. Appl., 4(4):Paper No. S2R26, 6,

work page doi:10.1515/jgth.2003.020 2003

[12] [12]

[HS24] Brian Hopkins and James A

doi:10.54550/eca2024v4s4r26. [HS24] Brian Hopkins and James A. Sellers. On Blecher and Knopfmacher’s fixed points for integer partitions. Discrete Math., 347(5):Paper No. 113938, 9,

work page doi:10.54550/eca2024v4s4r26

[13] [13]

[NP98] Peter M

doi:10.1016/j.disc.2024.113938. [NP98] Peter M. Neumann and Cheryl E. Praeger. Derangements and eigenvalue-free elements in finite classical groups. J. London Math. Soc. (2) , 58(3):564–586,

work page doi:10.1016/j.disc.2024.113938 2024

[14] [14]

[Pet] Fedor Petrov

doi:10.1112/S0024610798006772. [Pet] Fedor Petrov. A bijection between pairs of partitions. MathOverflow. URL: https://mathoverflow.net/ q/487338. [Ser03] Jean-Pierre Serre. On a theorem of Jordan. Bull. Amer. Math. Soc. (N.S.) , 40(4):429–440,

work page doi:10.1112/s0024610798006772

[15] [15]

[Spi17] Pablo Spiga

doi: 10.1090/S0273-0979-03-00992-3 . [Spi17] Pablo Spiga. On the number of derangements and derangements of prime power order of the affine general linear groups. J. Algebraic Combin., 45(2):345–362,

work page doi:10.1090/s0273-0979-03-00992-3

[16] [16]

[Ste68] Robert Steinberg

doi:10.1007/s10801-016-0709-3 . [Ste68] Robert Steinberg. Endomorphisms of linear algebraic groups, volume No. 80 of Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI,

work page doi:10.1007/s10801-016-0709-3