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arxiv: 1907.02396 · v1 · pith:TI4SZETKnew · submitted 2019-07-04 · 🧮 math.GR

Exponent of a finite group admitting a coprime automorphism

Sara Rodrigues , Pavel Shumyatsky This is my paper

Pith reviewed 2026-05-25 08:47 UTC · model grok-4.3

classification 🧮 math.GR
keywords finite groupscoprime automorphismsgroup exponentcentralizersautomorphism actionssoluble subgroupsnilpotent groupsderived length
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The pith

If every element of the centralizer and G_{-φ} lies in a φ-invariant subgroup of exponent dividing e, then the exponent of a finite group with coprime automorphism of order n is (e,n)-bounded.

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

The paper proves two theorems on exponents in finite groups G that admit a coprime automorphism φ of order n. The first shows that when every element of the centralizer G_φ and the set G_{-φ} belongs to some φ-invariant subgroup whose exponent divides e, the exponent of G itself is bounded by a number depending only on e and n. The second shows that if G_φ is nilpotent of class c, every element of G_{-φ} has order dividing e, and any two elements of G_{-φ} lie in a φ-invariant soluble subgroup of derived length d, then the exponent of the commutator subgroup [G,φ] is bounded by a number depending only on c, d, e and n. These statements give explicit conditions under which the action of a coprime automorphism forces the group (or a key subgroup) to have bounded exponent.

Core claim

Let G be a finite group admitting a coprime automorphism φ of order n. If every element from G_φ ∪ G_{-φ} is contained in a φ-invariant subgroup of exponent dividing e, then the exponent of G is (e,n)-bounded. Suppose that G_φ is nilpotent of class c. If x^e=1 for each x ∈ G_{-φ} and any two elements of G_{-φ} are contained in a φ-invariant soluble subgroup of derived length d, then the exponent of [G,φ] is bounded in terms of c,d,e,n.

What carries the argument

The sets G_φ (centralizer of φ) and G_{-φ} = {x^{-1}x^φ | x ∈ G}, together with the φ-invariant subgroups that contain their elements and control exponent propagation under the coprimeness condition on n.

If this is right

  • The exponent of G is bounded by a function of e and n alone under the first set of hypotheses.
  • The exponent of [G,φ] is bounded by a function of c, d, e and n alone under the second set of hypotheses.
  • The bounds are independent of the order of G.
  • The conclusions apply uniformly to any finite group admitting such an automorphism.

Where Pith is reading between the lines

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

  • The results supply a route to proving that certain classes of groups with coprime automorphisms must themselves have bounded exponent.
  • The second theorem indicates that nilpotency of the centralizer plus pairwise solubility conditions on G_{-φ} suffice to control the exponent of the derived subgroup generated by the automorphism action.
  • The techniques may extend to questions about the derived length or Fitting height of groups satisfying similar local conditions.

Load-bearing premise

Every element from G_φ ∪ G_{-φ} is contained in a φ-invariant subgroup of exponent dividing e (or the analogous nilpotency, order, and solubility conditions in the second theorem).

What would settle it

A finite group G with coprime automorphism φ of order n in which all elements of G_φ and G_{-φ} lie in φ-invariant subgroups of exponent dividing e, yet the exponent of G grows without bound as |G| increases.

read the original abstract

Let $G$ be a finite group admitting a coprime automorphism $\phi$ of order $n$. Denote by $G_{\phi}$ the centralizer of $\phi$ in $G$ and by $G_{-\phi}$ the set $\{ x^{-1}x^{\phi}; \ x\in G\}$. We prove the following results. 1. If every element from $G_{\phi}\cup G_{-\phi}$ is contained in a $\phi$-invariant subgroup of exponent dividing $e$, then the exponent of $G$ is $(e,n)$-bounded. 2. Suppose that $G_{\phi}$ is nilpotent of class $c$. If $x^{e}=1$ for each $x \in G_{-\phi}$ and any two elements of $G_{-\phi}$ are contained in a $\phi$-invariant soluble subgroup of derived length $d$, then the exponent of $[G,\phi]$ is bounded in terms of $c,d,e,n$.

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

0 major / 2 minor

Summary. The paper proves two conditional exponent bounds for a finite group G admitting a coprime automorphism φ of order n. First: if every element of G_φ ∪ G_{-φ} lies in some φ-invariant subgroup of exponent dividing e, then exp(G) is (e,n)-bounded. Second: if G_φ is nilpotent of class c, every element of G_{-φ} satisfies x^e = 1, and any two elements of G_{-φ} lie in a φ-invariant soluble subgroup of derived length d, then exp([G,φ]) is bounded in terms of c,d,e,n.

Significance. If the proofs are correct, the results supply useful conditional bounds in the theory of coprime automorphisms of finite groups, extending classical work on exponent restrictions under automorphism actions. The hypotheses are stated explicitly and enter the conclusions directly, which is a strength for applicability in inductive arguments or local-to-global exponent problems.

minor comments (2)
  1. The definition of G_{-φ} as {x^{-1}x^φ : x ∈ G} is standard but would benefit from an explicit reminder in §1 that this set is not necessarily a subgroup.
  2. Notation for the (e,n)-boundedness in Theorem 1 could be expanded in the introduction to clarify that the bound depends only on e and n (and not on |G| or other parameters).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation to accept the manuscript. The report accurately summarizes the two main results on conditional exponent bounds for finite groups admitting coprime automorphisms.

Circularity Check

0 steps flagged

No circularity; conditional theorems derived from explicit hypotheses

full rationale

The paper states two conditional results in the abstract and proves them under explicitly listed assumptions on G_φ and G_{-φ} (nilpotency class, exponent bounds, solubility conditions, and φ-invariance). These premises enter the conclusions directly as hypotheses, with no reduction of the claimed bounds to fitted parameters, self-definitional loops, or load-bearing self-citations that themselves presuppose the target exponent bounds. The derivation is a standard proof in finite group theory relying on coprime automorphism properties and subgroup constructions; no step equates a prediction to its input by construction or renames a known result as a new derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from finite group theory such as the existence and properties of centralizers, commutators, nilpotency, and solubility in groups. No free parameters or invented entities are apparent from the abstract.

axioms (2)
  • domain assumption G is a finite group admitting a coprime automorphism phi of order n
    Core setup stated in the abstract for both results.
  • standard math Standard properties of centralizers, derived subgroups, and phi-invariance hold in finite groups
    Invoked implicitly throughout the statements of the theorems.

pith-pipeline@v0.9.0 · 5697 in / 1423 out tokens · 37288 ms · 2026-05-25T08:47:31.521038+00:00 · methodology

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