A Paper on Calculating the Height and Relational Complexity of the Primitive Actions of $ PSL_{2} (q) $ and $ PGL_{2} (q) $

Scott Hudson

REVIEW 2 major objections 5 minor 25 references

Theorems 1.0.1 and 1.0.2 give exact height and relational complexity for every primitive action of PSL(2,q) and PGL(2,q) on maximal subgroups, for q≥11.

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2026-08-01 10:15 UTC pith:5IXSACBK

load-bearing objection Solid Borel/dihedral/A4 material, but the A5 upper bound rests on a GAP-assisted classification whose hand-fix doesn't close the conjugacy loophole. the 2 major comments →

arxiv 2607.20295 v1 pith:5IXSACBK submitted 2026-07-22 math.GR

A Paper on Calculating the Height and Relational Complexity of the Primitive Actions of PSL₂ (q) and PGL₂ (q)

Scott Hudson This is my paper

classification math.GR MSC 20B0520G40

keywords relational complexityheight of group actionprimitive actionsPSL(2,q)PGL(2,q)almost independent setsdihedral subgroupsSuzuki groups

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper determines, for essentially all finite fields, the two statistics—height and relational complexity—that measure how complex a group action is. Its main theorems give exact values for every primitive action of PSL(2,q) and PGL(2,q) on cosets of a maximal subgroup, with only the S4 action's relational complexity left as a two-element range. These numbers are the group-theoretic shadows of quantifier complexity in the corresponding structures. The paper also introduces almost independent sets as a general tool and exports the dihedral analysis to Suzuki groups.

Core claim

The paper proves that for q≥11, every primitive action of PSL(2,q) and PGL(2,q) on cosets of a maximal subgroup has height and relational complexity given by a short table: Borel (3,4); dihedral (3,3); A4 (2,3); S4 (3,3 or 4); A5 (3,4); subfield PSL(2,3)-type (2,3); other subfield (3,4); and the PGL index-2 PSL action (1,2). The S4 relational complexity is proved only to lie in {3,4}.

What carries the argument

The engine is the inequality RC(G,Ω) ≤ Ht(G,Ω)+1, which lets the author pin relational complexity by first bounding height through independent sets in stabilizer chains. A new notion, almost independent sets (sets that are not independent but all proper subsets are independent), decides when the +1 is actually attained. The dihedral cases are resolved by counting conjugates of a maximal dihedral subgroup and their pairwise intersections; the A4, S4, A5 and subfield cases reduce to classifying which subgroups can appear as intersections of two or three point stabilizers, sometimes by computer-assisted enumeration of possible stabilizer configurations.

Load-bearing premise

The S4 and A5 height/RC upper bounds rest on computer-assisted classification lemmas (7.3.5, 8.2.7, 8.2.10) that enumerate possible stabilizer configurations for hypothetical independent sets of size 4; the paper admits the code does not track conjugacy classes, producing spurious candidates later excluded by hand, and the load-bearing premise is that those exclusions are complete and the search has no false negatives.

What would settle it

For a prime p>7, find an independent set of size 4 in the S4 action of PSL(2,p) (equivalently, four maximal S4 subgroups no three of which have the same point stabilizer after intersecting). The paper's height-3 claim for all p≠5 requires that such a set does not exist; exhibiting one would refute Theorem 1.0.1's S4 row. Similarly, exhibiting an independent set of size 4 in the A5 action of PSL(2,q) for any q≥11 would refute the A5 row.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

For every q≥11 the two families have a complete, finite table of heights and relational complexities, settling these statistics for all primitive actions.
The bound RC ≤ Ht+1 is shown to be sharp: the +1 occurs exactly when almost independent sets of size Ht+1 exist.
The dihedral machinery extends beyond PSL2(q)/PGL2(q): the same argument proves the Suzuki groups Sz(q) acting on maximal dihedral subgroups have relational complexity 3.
The S4 case is the only unresolved cell; the paper conjectures RC=4 for PGL2(p), p>7, and shows RC=3 for PSL2(7), PSL2(17) and PSL2(47).

Where Pith is reading between the lines

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

If the conjecture on S4 actions of PGL2(p) holds, the value 3 occurs in PSL2 only at p≡±17 (mod 64) up to p=79, suggesting a congruence pattern that could be tested for larger primes.
Almost independent sets may serve as a practical certificate in other families: to show RC is one more than height it suffices to exhibit such a set, and to show equality it suffices to rule them out; this could be automated.
The technique of excluding spurious computer-generated candidates by hand suggests that a fully conjugacy-aware search would either confirm the S4/A5 bounds or reveal a missing configuration; the paper's explicit acknowledgment of this gap makes it the natural next check.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Solid Borel/dihedral/A4 material, but the A5 upper bound rests on a GAP-assisted classification whose hand-fix doesn't close the conjugacy loophole.

read the letter

Two things to know about this paper. It computes exact height and relational complexity for most primitive actions of PSL2(q) and PGL2(q), with honest caveats, and it introduces a genuinely useful tool (almost independent sets) that pays off in the dihedral chapter. But the A5 upper bound is not airtight: it rests on a GAP-assisted classification with an admitted conjugacy blind spot, and the hand-exclusion of the spurious PSL2(11) candidate in Lemma 8.2.6 does not go through as written.

The Borel and dihedral chapters are the strong part. The proofs are detailed human case work; I checked the main constructions (the I,J pair in the Borel theorem, the intersection-counting in the dihedral lemmas) and found them internally consistent. The dihedral theorems are stated generally and exported to Suzuki groups, which is a nice new result. The paper is also careful about marking what is review material and what is new, and it leaves the S4 relational complexity as '3 or 4' plus a conjecture rather than pretending to resolve it.

The soft spots are real. The S4 upper bound depends on a GAP enumeration that produced a spurious PSL2(7); that one is excluded using the published tables, which is acceptable. The A5 case is more serious. Lemma 8.2.6 needs to exclude PSL2(11) by hand. The proof picks g in G with g^{-1}H1g=H3 and g^{-1}H13g=H13, then cites Lemma 8.2.3 to say A4 subgroups of PSL2(11) are self-normalizing, so g lies in H13. But Lemma 8.2.3's self-normalizing statement is for the ambient group PSL2(11) itself; when the ambient G has q ≡ ±1 mod 8, N_G(H13) ≅ S4, so g need not be in H13. The contradiction does not follow. Since Lemma 8.2.7, Corollary 8.2.8 and Lemma 8.2.10 all lean on this, the A5 row (Ht=3, RC=4) is not established. It may be repairable with a sharper argument or a GAP run that tracks conjugacy, but as it stands it is a load-bearing gap. The subfield chapter I could not inspect beyond its preliminaries, so that row is less independently verified.

Who should read this: people working on height/relational complexity of finite permutation groups. The Borel, dihedral and A4 material is likely solid and usable. I would not cite the A5 result until it is fixed. It does deserve a serious referee — send it out — but the referee should focus on the A5 upper bound and ask for a repaired proof before acceptance.

Referee Report

2 major / 5 minor

Summary. The paper computes, for q ≥ 11, the height and relational complexity of every primitive action of PSL(2,q) and PGL(2,q) on cosets of a maximal subgroup. The main results are Theorems 1.0.1 and 1.0.2: Borel (3,4), dihedral (3,3), A4 (2,3), S4 (3,3 or 4), A5 (3,4), subfield (2,3 or 3,4), and for PGL(2,q) the index-2 PSL(2,q) action (1,2). The proof combines general results on height and relational complexity, a new notion of almost independent sets, detailed case analyses for the Borel and dihedral actions, and GAP-assisted classification arguments for the S4 and A5 actions. The paper also exports the dihedral machinery to Suzuki groups (Theorem 5.4.7).

Significance. If the results are correct, the paper gives exact asymptotic statistics for an infinite family of classical groups, introduces almost independent sets as a new tool, and provides a general theorem for dihedral subgroups that extends beyond PSL(2,q) and PGL(2,q). The explicit GAP code in Appendices A–D is a strength, and the small cases are checked against independent tables. However, the S4 and A5 upper bounds rest on computer-assisted classification lemmas whose completeness is not fully certified, and one step in the A5 proof contains a normalizer gap that is load-bearing for the exact A5 row of the main theorem.

major comments (2)

[§8.2, Lemma 8.2.6] The proof of Lemma 8.2.6 contains a gap that is load-bearing for the A5 row of Theorem 1.0.1. After GAP shows that a certain configuration generates a group L≅PSL2(11), the proof chooses g∈G with g^{-1}H1g=H3 and g^{-1}H13g=H13, then cites Lemma 8.2.3 to conclude that A4 subgroups of PSL2(11) are self-normalizing, hence g∈H13. But Lemma 8.2.3 concerns normalizers inside PSL2(q), not inside the generated subgroup L, and when q≡±1 (mod 8) the normalizer of an A4 in G is S4. The element g is only known to lie in G; it is not shown to lie in L. Thus the contradiction as written applies only when N_G(H13)=H13, e.g. when the ambient group is itself PSL2(11), not for all q satisfying the A5 hypotheses. This is not a local issue: Lemma 8.2.6 is used to prove Corollary 8.2.8, Lemma 8.2.9, Lemma 8.2.10, and hence Ht=3 and RC=4 for the A5 action. Please justify that the conjugating element can be c
[§7.3, Lemma 7.3.5; §8.2, Lemma 8.2.10] The height upper bounds for the S4 and A5 actions depend on GAP-assisted classification lemmas whose completeness is not established in the text. For example, Lemma 7.3.5 says a possible independent 4-set forces one of four alternatives and refers to Appendix A; Lemma 8.2.10 similarly depends on Appendix D. The paper explicitly admits that the GAP code does not track conjugacy classes and that this produced spurious candidate groups (PSL2(7) in Chapter 7 and PSL2(11) in Chapter 8), which are then excluded by hand. Excluding a false positive that arises from a known blind spot is not the same as proving that the enumeration has no false negatives. If an unenumerated configuration exists, the S4/A5 height bounds could fail. For a machine-assisted proof, the code and output need to certify the search space, or the relevant classification needs a human proof.

minor comments (5)

[Introduction, p.5] Typo: 'In ultimate goal of this paper' should be 'The ultimate goal of this paper'.
[Notation, §1] The convention for δ in Theorem 5.3.22 and Chapter 9 is introduced piecemeal; a central table of δ, δ_m and the corresponding dihedral/cyclic orders would improve readability.
[§5.4, Lemma 5.4.1] The notation Send_{G_{I_1}}(I_4,J_4) is used freely; it is defined in §5.1, but a reminder near first use in the proof of Lemma 5.4.1 would help.
[Tables 3.2.7–3.2.9] The completeness of the maximal subgroup classification is imported from [1] and [13]. The paper says this is not new research, but the exact list of primitive actions depends on it; a sentence explicitly stating which theorem in [1]/[13] supplies each table would be useful.
[Chapter 7, Conjecture 7.4.2] The discussion of p≡±17 (mod 64) and the exclusion of p=239 is clear, but it would be better to separate the verified computation for PSL2(79) from the speculative pattern, since the latter is not used in the proofs.

Circularity Check

0 steps flagged

No significant circularity: the paper's height and relational complexity results are derived from group structure by case analysis, with external benchmarks used only for small cases and lower bounds.

full rationale

The paper's central claims (Theorems 1.0.1 and 1.0.2) are exact statistics computed for each primitive action of PSL(2,q) and PGL(2,q) using the maximal subgroup classifications imported from [1] and [13], the algebraic structure of the groups, and independent-set arguments. No quantity used in the derivations is defined in terms of the quantity being derived, and no parameter is fitted to a target value. The almost independent set notion (Definition 2.4.3) is a new analytic tool, not a disguised restatement of the desired result. Small-q values are taken from published tables [25] and lower bounds such as RC ≥ 3 from [9]; these are external, independently checkable benchmarks rather than author self-citations or fitted inputs. The GAP-assisted classification lemmas (7.3.5, 8.2.7, 8.2.10) do contain admitted limitations: the code does not track conjugacy classes, producing spurious PSL(2,7) and PSL(2,11) candidates that are excluded by hand. That is a correctness risk about completeness of the enumeration, but it is not circular: the excluded candidates are not equivalent to assuming the theorem, and the central derivation does not reduce to its own inputs. There are no load-bearing self-citations: the author does not rely on prior work of the same author as the justification for the main results. Accordingly, no specific circular step can be exhibited, and the appropriate finding is no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters: nothing is fitted, tuned, or chosen ad hoc to force a table entry; the statistics are computed from group structure. The contributions sit on three imports: the maximal-subgroup classification (external), the small-case tables (external, [25]), and the paper's own GAP computations, whose exhaustiveness is assumed rather than proved — this last is the main audit charge. Standard PSL(2,q) background is routine. No new physical/mathematical entities are postulated; 'almost independent sets' is a definitional screening tool.

axioms (4)

domain assumption Classification of maximal subgroups of PSL(2,q) and PGL(2,q) (Tables 3.2.7–3.2.9, from [13] Thm 3.5 and [1] pp. 377/380) is complete.
Theorems 1.0.1/1.0.2 enumerate actions by maximal subgroup type; a missing type would silently drop a primitive action. Small-q exceptions (q = 5,7,9,11) are treated separately, with checks against [25].
domain assumption GAP computations in Appendices A–D and in Lemmas 7.3.5, 8.2.7, 8.2.10 are correct and exhaustive modulo the paper's manual conjugacy exclusions.
The height upper bound for S4 actions and the height/RC upper bounds for A5 actions are shown 'in Appendix A/D' and by embedded GAP presentations. The paper admits code blind spots (spurious PSL(2,7) and PSL(2,11)), so completeness rests on human exclusions.
domain assumption Small-case statistics taken from the tables in [25] (e.g., q = 3,5 Borel RC; PSL(2,4)/PSL(2,5) on D6 with Ht = 2; PGL(2,q) dihedral for q in {7,9,11}; PSL(2,7) S4 RC = 3).
Chapters 4–8 repeatedly dispatch exceptional q-values to [25]; the general theorems inherit these externally computed values.
standard math Standard structural facts about PSL(2,q)/PGL(2,q): orders (Lemma 3.2.2), simplicity, involution conjugacy (Lemmas 3.3.4, 3.3.5), Sylow theorems, and the model-theoretic arity background (Cherlin [2],[3]).
Routine background; proofs or standard references are cited (e.g., [22] p. 19, [14] Lemma A.3, [18], [7]).

pith-pipeline@v1.3.0-alltime-deepseek · 78551 in / 29750 out tokens · 240109 ms · 2026-08-01T10:15:24.814848+00:00 · methodology

0 comments

read the original abstract

For a finite group acting on a finite set, a statistic called relational complexity can be calculated for the action. This notion was defined by Gregory Cherlin and motivated by considerations in model theory. Another related statistic is the height of the action, which provides an upper bound for relational complexity. In this paper, both concepts are defined and some basic results proved. The main focus later on is examining the primitive actions of $ PSL_{2} (q) $ and $ PGL_{2} (q) $ and computing both the height and relational complexity for each one.

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 1 linked inside Pith

[1]

Bray, Derek F

John N. Bray, Derek F. Holt, and Colva M. Roney-Dougal.The maximal subgroups of the low-dimensional finite classical groups., volume 407. Cambridge: Cambridge University Press, 2013

2013
[2]

Sporadic homogeneous structures

Gregory Cherlin. Sporadic homogeneous structures. InThe Gelfand Mathematical Seminars, 1996–1999. Dedicated to the memory of Chih-Han Sah, pages 15–48. Boston, MA: Birkh¨ auser, 2000

1996
[3]

On the relational complexity of a finite permutation group.J

Gregory Cherlin. On the relational complexity of a finite permutation group.J. Algebr. Comb., 43(2):339–374, 2016

2016
[4]

Cherlin, Gary A

Gregory L. Cherlin, Gary A. Martin, and Daniel H. Saracino. Arities of permutation groups: Wreath products andk-sets.J. Comb. Theory, Ser. A, 74(2):249–286, 1996

1996
[5]

H. S. M. Coxeter and W. O. J. Moser.Generators and relations for discrete groups. 3rd ed., volume 14. Springer-Verlag, Berlin, 1972

1972
[6]

Cherlin’s conjecture for sporadic simple groups.Pac

Francesca Dalla Volta, Nick Gill, and Pablo Spiga. Cherlin’s conjecture for sporadic simple groups.Pac. J. Math., 297(1):47–66, 2018

2018
[7]

Dixon and Brian Mortimer.Permutation groups., volume 163

John D. Dixon and Brian Mortimer.Permutation groups., volume 163. New York, NY: Springer-Verlag, 1996

1996
[8]

The GAP Group.GAP – Groups, Algorithms, and Programming, Version 4.11.1, 2021

2021
[9]

Cherlin’s conjecture for almost simple groups of lie rank 1.Mathe- matical Proceedings of the Cambridge Philosophical Society, 167(3):417–435, Jul 2018

Nick Gill, Francis Hunt, and Pablo Spiga. Cherlin’s conjecture for almost simple groups of lie rank 1.Mathe- matical Proceedings of the Cambridge Philosophical Society, 167(3):417–435, Jul 2018

2018
[10]

Liebeck, and Pablo Spiga.Cherlin ’s conjecture for finite primitive binary permutation groups, volume 2302 ofLect

Nick Gill, Martin W. Liebeck, and Pablo Spiga.Cherlin ’s conjecture for finite primitive binary permutation groups, volume 2302 ofLect. Notes Math.Cham: Springer, 2022

2022
[11]

Statistics for sn acting on k-sets.Journal of Algebra, 607:286–299, 2022

Nick Gill and Bianca Lod` a. Statistics for sn acting on k-sets.Journal of Algebra, 607:286–299, 2022

2022
[12]

On the height and relational complexity of a finite permutation group.Nagoya Mathematical Journal, 246:372–411, 2022

Nick Gill, Bianca Lod´ a, and Pablo Spiga. On the height and relational complexity of a finite permutation group.Nagoya Mathematical Journal, 246:372–411, 2022

2022
[13]

Maximal subgroups of almost simple groups with socle PSL(2,q).arXiv Mathematics e-prints, page math/0703685, March 2007

Michael Giudici. Maximal subgroups of almost simple groups with socle PSL(2,q).arXiv Mathematics e-prints, page math/0703685, March 2007

Pith/arXiv arXiv 2007
[14]

Guralnick and Michael E

Robert M. Guralnick and Michael E. Zieve. Polynomials with PSL(2) monodromy.Ann. Math. (2), 172(2):1315– 1359, 2010

2010
[15]

I, volume 134

Bertram Huppert.Endliche Gruppen. I, volume 134. Springer, Cham, 1967

1967
[16]

D. L. Johnson.Presentations of Groups. London Mathematical Society Student Texts. Cambridge University Press, 2 edition, 1997. 128

1997
[17]

On relational complexity and base size of finite primitive groups

Veronica Kelsey and Colva M Roney-Dougal. On relational complexity and base size of finite primitive groups. Pacific Journal of Mathematics, 318(1):89–108, 2022

2022
[18]

Rose.A course on group theory

John S. Rose.A course on group theory. Repr. of the 1978 orig.New York, NY: Dover Publications, repr. of the 1978 orig. edition, 1994

1978
[19]

Rotman.An introduction to the theory of groups., volume 148

Joseph J. Rotman.An introduction to the theory of groups., volume 148. New York, NY: Springer-Verlag, 1995

1995
[20]

A new type of simple groups of finite order.Proc

Michio Suzuki. A new type of simple groups of finite order.Proc. Natl. Acad. Sci. USA, 46:868–870, 1960

1960
[21]

On a class of doubly transitive groups.Ann

Michio Suzuki. On a class of doubly transitive groups.Ann. Math. (2), 75:105–145, 1962

1962
[22]

Taylor.The geometry of the classical groups., volume 9

Donald E. Taylor.The geometry of the classical groups., volume 9. Berlin: Heldermann Verlag, 1992

1992
[23]

A reduction theorem for primitive binary permutation groups.Bulletin of the London Mathematical Society, 48(2):291–299, 2016

Joshua Wiscons. A reduction theorem for primitive binary permutation groups.Bulletin of the London Mathematical Society, 48(2):291–299, 2016

2016
[24]

Joshua Wiscons’ Research Page.http://webpages.csus.edu/wiscons/research/, 2022

Joshua Wiscons. Joshua Wiscons’ Research Page.http://webpages.csus.edu/wiscons/research/, 2022. [Online; accessed 19-April-2022]

2022
[25]

Tables for Height and Relational Complexity of Primitive Actions of Degree at Most 100.http://webpages.csus.edu/wiscons/research/code/RCompDataLatex.pdf, 2022

Joshua Wiscons. Tables for Height and Relational Complexity of Primitive Actions of Degree at Most 100.http://webpages.csus.edu/wiscons/research/code/RCompDataLatex.pdf, 2022. [Online; accessed 19-April-2022]. 129

2022

[1] [1]

Bray, Derek F

John N. Bray, Derek F. Holt, and Colva M. Roney-Dougal.The maximal subgroups of the low-dimensional finite classical groups., volume 407. Cambridge: Cambridge University Press, 2013

2013

[2] [2]

Sporadic homogeneous structures

Gregory Cherlin. Sporadic homogeneous structures. InThe Gelfand Mathematical Seminars, 1996–1999. Dedicated to the memory of Chih-Han Sah, pages 15–48. Boston, MA: Birkh¨ auser, 2000

1996

[3] [3]

On the relational complexity of a finite permutation group.J

Gregory Cherlin. On the relational complexity of a finite permutation group.J. Algebr. Comb., 43(2):339–374, 2016

2016

[4] [4]

Cherlin, Gary A

Gregory L. Cherlin, Gary A. Martin, and Daniel H. Saracino. Arities of permutation groups: Wreath products andk-sets.J. Comb. Theory, Ser. A, 74(2):249–286, 1996

1996

[5] [5]

H. S. M. Coxeter and W. O. J. Moser.Generators and relations for discrete groups. 3rd ed., volume 14. Springer-Verlag, Berlin, 1972

1972

[6] [6]

Cherlin’s conjecture for sporadic simple groups.Pac

Francesca Dalla Volta, Nick Gill, and Pablo Spiga. Cherlin’s conjecture for sporadic simple groups.Pac. J. Math., 297(1):47–66, 2018

2018

[7] [7]

Dixon and Brian Mortimer.Permutation groups., volume 163

John D. Dixon and Brian Mortimer.Permutation groups., volume 163. New York, NY: Springer-Verlag, 1996

1996

[8] [8]

The GAP Group.GAP – Groups, Algorithms, and Programming, Version 4.11.1, 2021

2021

[9] [9]

Cherlin’s conjecture for almost simple groups of lie rank 1.Mathe- matical Proceedings of the Cambridge Philosophical Society, 167(3):417–435, Jul 2018

Nick Gill, Francis Hunt, and Pablo Spiga. Cherlin’s conjecture for almost simple groups of lie rank 1.Mathe- matical Proceedings of the Cambridge Philosophical Society, 167(3):417–435, Jul 2018

2018

[10] [10]

Liebeck, and Pablo Spiga.Cherlin ’s conjecture for finite primitive binary permutation groups, volume 2302 ofLect

Nick Gill, Martin W. Liebeck, and Pablo Spiga.Cherlin ’s conjecture for finite primitive binary permutation groups, volume 2302 ofLect. Notes Math.Cham: Springer, 2022

2022

[11] [11]

Statistics for sn acting on k-sets.Journal of Algebra, 607:286–299, 2022

Nick Gill and Bianca Lod` a. Statistics for sn acting on k-sets.Journal of Algebra, 607:286–299, 2022

2022

[12] [12]

On the height and relational complexity of a finite permutation group.Nagoya Mathematical Journal, 246:372–411, 2022

Nick Gill, Bianca Lod´ a, and Pablo Spiga. On the height and relational complexity of a finite permutation group.Nagoya Mathematical Journal, 246:372–411, 2022

2022

[13] [13]

Maximal subgroups of almost simple groups with socle PSL(2,q).arXiv Mathematics e-prints, page math/0703685, March 2007

Michael Giudici. Maximal subgroups of almost simple groups with socle PSL(2,q).arXiv Mathematics e-prints, page math/0703685, March 2007

Pith/arXiv arXiv 2007

[14] [14]

Guralnick and Michael E

Robert M. Guralnick and Michael E. Zieve. Polynomials with PSL(2) monodromy.Ann. Math. (2), 172(2):1315– 1359, 2010

2010

[15] [15]

I, volume 134

Bertram Huppert.Endliche Gruppen. I, volume 134. Springer, Cham, 1967

1967

[16] [16]

D. L. Johnson.Presentations of Groups. London Mathematical Society Student Texts. Cambridge University Press, 2 edition, 1997. 128

1997

[17] [17]

On relational complexity and base size of finite primitive groups

Veronica Kelsey and Colva M Roney-Dougal. On relational complexity and base size of finite primitive groups. Pacific Journal of Mathematics, 318(1):89–108, 2022

2022

[18] [18]

Rose.A course on group theory

John S. Rose.A course on group theory. Repr. of the 1978 orig.New York, NY: Dover Publications, repr. of the 1978 orig. edition, 1994

1978

[19] [19]

Rotman.An introduction to the theory of groups., volume 148

Joseph J. Rotman.An introduction to the theory of groups., volume 148. New York, NY: Springer-Verlag, 1995

1995

[20] [20]

A new type of simple groups of finite order.Proc

Michio Suzuki. A new type of simple groups of finite order.Proc. Natl. Acad. Sci. USA, 46:868–870, 1960

1960

[21] [21]

On a class of doubly transitive groups.Ann

Michio Suzuki. On a class of doubly transitive groups.Ann. Math. (2), 75:105–145, 1962

1962

[22] [22]

Taylor.The geometry of the classical groups., volume 9

Donald E. Taylor.The geometry of the classical groups., volume 9. Berlin: Heldermann Verlag, 1992

1992

[23] [23]

A reduction theorem for primitive binary permutation groups.Bulletin of the London Mathematical Society, 48(2):291–299, 2016

Joshua Wiscons. A reduction theorem for primitive binary permutation groups.Bulletin of the London Mathematical Society, 48(2):291–299, 2016

2016

[24] [24]

Joshua Wiscons’ Research Page.http://webpages.csus.edu/wiscons/research/, 2022

Joshua Wiscons. Joshua Wiscons’ Research Page.http://webpages.csus.edu/wiscons/research/, 2022. [Online; accessed 19-April-2022]

2022

[25] [25]

Tables for Height and Relational Complexity of Primitive Actions of Degree at Most 100.http://webpages.csus.edu/wiscons/research/code/RCompDataLatex.pdf, 2022

Joshua Wiscons. Tables for Height and Relational Complexity of Primitive Actions of Degree at Most 100.http://webpages.csus.edu/wiscons/research/code/RCompDataLatex.pdf, 2022. [Online; accessed 19-April-2022]. 129

2022