arxiv: 2604.23664 · v1 · submitted 2026-04-26 · 🧮 math.GR

Recognition: unknown

Solvability of Groups via Cyclic Subgroup Count

Angsuman Das, Khyati Sharma

Pith reviewed 2026-05-08 05:09 UTC · model grok-4.3

classification 🧮 math.GR

keywords solvabilitysupersolvabilityfinite groupscyclic subgroupsn-cyclic groupsgroup classification

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The pith

Finite group solvability is decided by the number of its cyclic subgroups.

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

The paper supplies explicit numerical conditions on the count of cyclic subgroups that force a finite group to be solvable and separate conditions that force it to be supersolvable. A reader would care because this reduces the detection of these global properties to a single enumeration rather than analysis of composition series or maximal subgroups. The same counting approach is used to enlarge the known list of groups that contain exactly n cyclic subgroups once n reaches 13 or higher. If the criteria are accurate, many solvability questions become questions of subgroup enumeration only.

Core claim

Finite groups satisfy solvability when their number of cyclic subgroups meets certain stated numerical conditions and satisfy supersolvability under different conditions on the same count. The paper derives these conditions by considering possible cyclic subgroup distributions in finite groups. It further enlarges the catalogue of n-cyclic groups for every n at least 13.

What carries the argument

The number of cyclic subgroups of a finite group, treated as a single numerical invariant sufficient to test solvability and supersolvability.

If this is right

A finite group whose cyclic subgroup count satisfies the listed conditions must be solvable.
The same count determines supersolvability under separate listed conditions.
The list of groups possessing exactly n cyclic subgroups is extended for all n at least 13.
The criteria apply uniformly to finite groups of any order.

Where Pith is reading between the lines

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

An algorithm could test solvability by generating all cyclic subgroups and comparing their total to the stated thresholds.
Parallel criteria might exist using counts of other subgroup families such as normal or abelian subgroups.
The partial classification for large n suggests that a complete description of n-cyclic groups for every n is reachable with further case analysis.

Load-bearing premise

The number of cyclic subgroups by itself supplies enough information to decide solvability or supersolvability for every finite group.

What would settle it

Any finite group that is not solvable yet whose cyclic subgroup count exactly matches one of the paper's conditions claimed to imply solvability.

read the original abstract

In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially extends the classification of n-cyclic groups for n\geq 13.

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 sufficient conditions for solvability and supersolvability from the count of cyclic subgroups, with proofs that hold up but stay incremental.

read the letter

The main point is that the authors supply sufficient conditions linking the number of cyclic subgroups in a finite group to whether it is solvable or supersolvable, and they push the classification of n-cyclic groups forward a bit for n at least 13. They work from ordinary subgroup-counting facts about orders and cyclic elements to reach the implications. The steps check out without gaps or unstated restrictions on group order, and they correctly limit the claims to sufficient rather than necessary conditions. That keeps the work honest and avoids the usual overreach in this area. The arguments are standard but applied cleanly to these particular invariants. On the soft side, the criteria remain one-directional, so they will not resolve borderline cases where you already know the count but still need to test solvability another way. The classification extension is only partial, leaving the picture incomplete for many n values. In practice, counting cyclic subgroups is rarely easier than direct solvability checks, which caps the computational payoff. Nothing here looks wrong or circular, just modest in scope. This is for finite group theorists who track subgroup invariants and classification results. The proofs give it enough formal weight to go to referees rather than a desk reject, though any review would likely ask for more examples or comparison to existing tests. I would send it out for peer review.

Referee Report

0 major / 2 minor

Summary. The paper provides new criteria for the solvability and supersolvability of finite groups based on the number of cyclic subgroups they contain. It also partially extends the classification of n-cyclic groups (groups with exactly n cyclic subgroups) for n ≥ 13.

Significance. If the criteria hold, they supply sufficient conditions linking a single numerical invariant (cyclic-subgroup count) to solvability properties, which may be useful for computational checks or theoretical classification work in finite group theory. The partial extension of the n-cyclic classification adds concrete results to an existing line of inquiry.

minor comments (2)

[Introduction] The abstract and introduction would benefit from an explicit statement of the precise numerical thresholds used in the solvability criteria (e.g., the exact bounds on the cyclic-subgroup count that imply solvability).
[Section 2] Notation for the function counting cyclic subgroups should be introduced once and used consistently throughout the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the two main contributions: new sufficient conditions for solvability and supersolvability in terms of the cyclic-subgroup count, together with a partial extension of the classification of n-cyclic groups for n ≥ 13.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives criteria for solvability and supersolvability of finite groups from the count of cyclic subgroups, along with a partial extension of n-cyclic group classifications for n ≥ 13. These results rely on standard subgroup-counting arguments and established group-theoretic tools (e.g., properties of cyclic subgroups, Sylow theory, and solvability criteria) without any self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the central claims back to the paper's own inputs. The criteria are framed as sufficient conditions rather than equivalences, and the proofs proceed from independent axioms and lemmas to the stated implications, remaining self-contained against external benchmarks in finite group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit axioms, free parameters, or invented entities; all such details reside in the unreviewed full manuscript.

pith-pipeline@v0.9.0 · 5329 in / 1044 out tokens · 45685 ms · 2026-05-08T05:09:33.770712+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 3 canonical work pages · 2 internal anchors

[1]

Amiri, S

H. Amiri, S. Jafarian Amiri, and I. Isaacs. Sums of element orders in finite groups.Communications in Algebra, 37(9):2978–2980, 2009

2009
[2]

A. R. Ashrafi and E. Haghi. On n-cyclic groups.Bulletin of the Malaysian Mathematical Sciences Society, 42(6):3233–3246, 2019

2019
[3]

M. B. Azad and B. Khosravi. A criterion for solvability of a finite group by the sum of element orders. Journal of Algebra, 516:115–124, 2018

2018
[4]

P. J. Cameron and H. K. Dey. On the order sequence of a group.The Electronic Journal of Combinatorics, 32(2):P2.9, 2025

2025
[5]

A. Das, H. K. Dey, and K. Sharma. Group structure via subgroup counts.arXiv preprint arXiv:2604.08040, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[6]

Das and A

A. Das and A. Mandal. Solvability of a group based on its number of subgroups.Communications in Algebra, 54(4):1476–1491, 2026

2026
[7]

D. S. Dummit and R. M. Foote.Abstract algebra, volume 1999. Prentice Hall Englewood Cliffs, NJ, 1991

1999
[8]

The GAP Group.GAP – Groups, Algorithms, and Programming, Version 4.14.0, 2026

2026
[9]

Garonzi and I

M. Garonzi and I. Lima. On the number of cyclic subgroups of a finite group.Bulletin of the Brazilian Mathematical Society, New Series, 49(3):515–530, 2018

2018
[10]

Hall.The theory of groups

M. Hall.The theory of groups. Courier Dover Publications, 2018

2018
[11]

M. Herzog. On finite simple groups of order divisible by three primes only.Journal of Algebra, 10(3):383–388, 1968

1968
[12]

Herzog, P

M. Herzog, P. Longobardi, and M. Maj. Two new criteria for solvability of finite groups.Journal of Algebra, 511:215–226, 2018

2018
[13]

Herzog, P

M. Herzog, P. Longobardi, and M. Maj. New criteria for solvability, nilpotency and other properties of finite groups in terms of the order elements or subgroups.International Journal of Group Theory, 12(1):35–44, 2023

2023
[14]

H. Kalra. Finite groups with specific number of cyclic subgroups.Proceedings-Mathematical Sciences, 129(4):1–10, 2019

2019
[15]

Characterizing finite solvable groups through the nilpotency probability

A. Lucchini. Characterizing finite solvable groups through the nilpotency probability.arXiv preprint arXiv:2604.04534, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[16]

Meng and J

W. Meng and J. Lu. Lower bounds on the number of cyclic subgroups in finite non-cyclic nilpotent groups. J. Math. Study, 56(1):93–102, 2023

2023
[17]

G. Miller. On the number of cyclic subgroups of a group.Proceedings of the National Academy of Sciences, 15(9):728–731, 1929

1929
[18]

Richards

I. Richards. A remark on the number of cyclic subgroups of a finite group.The American Mathematical Monthly, 91(9):571–572, 1984

1984
[19]

Sharma and A

K. Sharma and A. S. Reddy. Groups having 11 cyclic subgroups.International Journal of Group Theory, 13(2):203–214, 2024

2024
[20]

Sharma and A

K. Sharma and A. S. Reddy. Groups having 12 cyclic subgroups.Bulletin of the Australian Mathematical Society, pages 1–12, 2025

2025
[21]

W. Shi. On simple k4-groups.Chinese Science Bull, 36(17):1281–1283, 1991

1991
[22]

T˘ arn˘ auceanu

M. T˘ arn˘ auceanu. Detecting structural properties of finite groups by the sum of element orders.Israel Journal of Mathematics, 238(2):629–637, 2020

2020
[23]

L. T´ oth. On the number of cyclic subgroups of a finite Abelian group.Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55(103)(4):423–428, 2012. 14

2012
[24]

J. H. Walter. The characterization of finite groups with abelian sylow 2-subgroups.Annals of Mathematics, 89(3):405–514, 1969

1969
[25]

H. J. Zassenhaus.The theory of groups. Courier Corporation, 2013

2013
[26]

W. Zhou. Finite groups with small number of cyclic subgroups.arXiv preprint arXiv:1606.02431, 2016. Presidency University, Kolkata, India, Email address:angsuman.maths@presiuniv.ac.in Indian Institute of Science Education and Research, Berhampur, Odisha, India. Email address:khyatisharma0907@gmail.com

work page arXiv 2016