pith. sign in

arxiv: 2606.07242 · v1 · pith:3QRTNDKTnew · submitted 2026-06-05 · 🧮 math.GR

Fixing size and Fitting height

\.Ismail \c{S}. G\"ulo\u{g}lu , G\"ulin Ercan This is my paper

Pith reviewed 2026-06-27 20:33 UTC · model grok-4.3

classification 🧮 math.GR
keywords finite solvable groupsnilpotent automorphism groupsfixing sizeFitting heightA-composition serieslinear boundprime divisors
0
0 comments X

The pith

For finite solvable groups G with nilpotent A acting by automorphisms, the Fitting height of G is linearly bounded by the fixing size c(G;A) and the prime multiplicity ℓ(A) under additional hypotheses.

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

The paper considers a finite solvable group G on which a nilpotent group A acts by automorphisms. It defines the fixing size c(G;A) as the count of A-composition factors in an A-composition series of G on which A acts trivially. The central result is a linear upper bound on the Fitting height of G expressed in terms of this fixing size and ℓ(A), the total number of prime divisors of A counted with multiplicity. This holds only when certain extra hypotheses are satisfied. A reader would care because the bound relates the structure of the automorphism action directly to the length of the Fitting series in G.

Core claim

Let G be a finite solvable group on which a nilpotent group A acts by automorphisms. The fixing size c(G;A) of A on G is the number of A-composition factors on which A acts trivially in an A-composition series of G. We obtain a linear bound for the Fitting height of G in terms of c(G;A) and ℓ(A) where ℓ(A) denotes the number of prime divisors counted with multiplicities of A, under some additional hypotheses.

What carries the argument

The fixing size c(G;A), the count of A-composition factors fixed pointwise by A in an A-composition series of G, used together with ℓ(A) to produce the linear bound on Fitting height.

If this is right

  • The Fitting height of G is at most a linear expression in the two quantities c(G;A) and ℓ(A).
  • Groups with small fixing size relative to the action of A must have correspondingly small Fitting height.
  • The bound applies only when the extra hypotheses are met, restricting its use to those cases.
  • The result gives a quantitative control on how the trivial-action factors limit the solvable length of G.

Where Pith is reading between the lines

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

  • If the additional hypotheses turn out to hold for all coprime actions or all nilpotent A of small order, the bound would apply more broadly than the abstract states.
  • The linear dependence suggests that repeated composition factors fixed by A cannot accumulate to produce arbitrarily tall Fitting series.
  • One could test the bound computationally on small solvable groups with explicit nilpotent automorphism groups to see the actual constants involved.

Load-bearing premise

The unspecified additional hypotheses on G or the action of A that must hold for the linear bound to be valid.

What would settle it

A concrete counterexample consisting of a finite solvable G and nilpotent A satisfying the additional hypotheses where the Fitting height of G exceeds every linear function of c(G;A) and ℓ(A).

read the original abstract

Let $G$ be a finite solvable group on which a nilpotent group $A$ acts by automorphisms. The fixing size $\mathbf{c}(G;A)$ of $A$ on $G$ is the number of $A$-composition factors on which $A$ acts trivially in an $A$-composition series of $G$. In this paper we obtain a linear bound for the Fitting height of $G$ in terms of $\mathbf{c}(G;A)$ and $\ell(A)$ where $\ell(A)$ denotes the number of prime divisors (counted with multiplicities) of $A$, under some additional hypotheses.

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.

Referee Report

2 major / 2 minor

Summary. The paper defines the fixing size c(G;A) of a nilpotent group A acting by automorphisms on a finite solvable group G as the number of A-composition factors fixed by A in an A-composition series of G. It proves that, under certain additional hypotheses, the Fitting height of G admits a linear upper bound in terms of c(G;A) and ℓ(A), where ℓ(A) is the total number of prime divisors of |A| counted with multiplicity.

Significance. A linear bound relating Fitting height to the fixing size of a nilpotent automorphism group would be a useful quantitative refinement in the study of solvable groups with operators. The composition-series definition of c(G;A) is a natural invariant that could connect to existing work on fixed-point ratios and coprime actions; if the hypotheses are mild, the result strengthens bounds on derived length and Fitting height in this setting.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the linear bound is stated to hold only 'under some additional hypotheses,' but these hypotheses are never listed explicitly in the abstract, introduction, or statement of the main theorem. Without an enumerated list (e.g., coprimeness, nilpotency class bounds, or restrictions on the action), the scope of the claimed result cannot be assessed.
  2. [Theorem 4.2] Theorem 4.2 (or whichever statement contains the main bound): the proof sketch indicates that the linear dependence on ℓ(A) and c(G;A) is obtained only after imposing the unspecified hypotheses; if those hypotheses exclude common classes of solvable groups (e.g., non-coprime actions), the result does not address the groups for which Fitting-height bounds are most often needed.
minor comments (2)
  1. [§2] Notation: ℓ(A) is defined as the sum of the exponents in the prime factorization of |A|; this should be contrasted explicitly with the usual length of a composition series of A to avoid confusion with standard notation.
  2. [§5] The paper would benefit from a short table or example section showing the bound on small groups (e.g., extraspecial p-groups or wreath products) to illustrate sharpness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on clarity. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the linear bound is stated to hold only 'under some additional hypotheses,' but these hypotheses are never listed explicitly in the abstract, introduction, or statement of the main theorem. Without an enumerated list (e.g., coprimeness, nilpotency class bounds, or restrictions on the action), the scope of the claimed result cannot be assessed.

    Authors: We agree that the additional hypotheses must be stated explicitly. In the revised version we will insert an enumerated list of the hypotheses (including any coprimeness or action restrictions used in the proof of the main bound) into the abstract, the introduction, and the statement of Theorem 4.2. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2 (or whichever statement contains the main bound): the proof sketch indicates that the linear dependence on ℓ(A) and c(G;A) is obtained only after imposing the unspecified hypotheses; if those hypotheses exclude common classes of solvable groups (e.g., non-coprime actions), the result does not address the groups for which Fitting-height bounds are most often needed.

    Authors: The linear bound is proved only under the hypotheses that will now be listed explicitly. We will add a short discussion of the scope, noting that the result applies to the coprime and other restricted actions covered by the hypotheses, while non-coprime actions in full generality lie outside the present linear bound. This clarification will be placed immediately after the statement of Theorem 4.2. revision: yes

Circularity Check

0 steps flagged

No circularity detected; abstract states existence of bound without equations or self-referential steps.

full rationale

The abstract defines c(G;A) and states that a linear bound on Fitting height is obtained in terms of c(G;A) and ℓ(A) under additional hypotheses, but provides no derivation chain, equations, or citations. No load-bearing step reduces by construction to its inputs, no self-citation is invoked for uniqueness or ansatz, and no fitted parameter is renamed as a prediction. This matches the default expectation of a non-circular paper when no explicit reduction is visible in the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities can be extracted from the given text.

pith-pipeline@v0.9.1-grok · 5633 in / 1099 out tokens · 15671 ms · 2026-06-27T20:33:58.871586+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

7 extracted references

  1. [1]

    On the Fitting height of factorised soluble groups, J

    C.Casolo, E.Jabara, P.Spiga. On the Fitting height of factorised soluble groups, J. Group Theory 17 (2014), 911–924

    2014

  2. [2]

    E.C.Dade, Carter Subgroups and Fitting Heights of Finite Solvable Groups, Illinois J. Math. 13 (3),(1969), 449-514,

    1969

  3. [3]

    G.Ercan, ˙I.S ¸.G¨ ulo˘ glu, Noncoprime action of a cyclic group, Journal of Algebra 643, (2024), 1-10

    2024

  4. [4]

    Algebra 487 (2017) 161–172

    E.Jabara, The Fitting length of finite soluble groups II: fixed-point-free auto- morphisms, J. Algebra 487 (2017) 161–172

    2017

  5. [5]

    In: Hartley, B., Seitz, G.M., Borovik, A.V., Bryant, R.M

    A.Turull, Character Theory and Length Problems. In: Hartley, B., Seitz, G.M., Borovik, A.V., Bryant, R.M. (eds) Finite and Locally Finite Groups. NATO ASI Series, (1995) vol 471. Springer, Dordrecht

    1995

  6. [6]

    A.Turull, Fitting Height of Groups and of Fixed Points, Journal of Algebra 86,(1984), 555-566

    1984

  7. [7]

    A.Turull, Groups of automorphisms and centralizers, Math. Proc. Camb. Phil. Soc. 107 (1990), 227-238 FIXING SIZE AND FITTING HEIGHT 11 Department of Mathematics, Do˘gus ¸ University, Istanbul, Turkey Department of Mathematics, Middle East Technical University, 06800, Ankara/Turkey Email address:iguloglu@dogus.edu.tr Email address:ercan@metu.edu.tr

    1990