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On a problem of B. Hartley about a small centralizer in finite and locally finite groups

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Pith's one-line read A finite group with an automorphism of order $n$ and $m$ fixed points always contains a soluble subgroup whose index and Fitting height are bounded by functions of $m$ and $n$.

desk verdict New arbitrary-order fixed-point theorem that settles Hartley's Kourovka problem; proof is sound apart from a localized, easily patched Hall-subgroup distinctness gap. read the letter →

arxiv 2505.20999 v2 pith:SIOSX55N submitted 2025-05-27 math.GR

Evgeny Khukhro This is my paper
classification math.GR MSC 20F4520D2520E3620E2520F5020F19
keywords finitegroupslocallyautomorphismcentralizerFittingheightnilpotentHartleyproblemfixed-point-free
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that a small fixed-point subgroup of an automorphism of a finite group forces the whole group to be close to soluble in a quantitative way. Specifically, if a finite group $G$ admits an automorphism $\varphi$ of order $n$ with $m$ fixed points, then $G$ has a soluble subgroup whose index and Fitting height are bounded by functions of $m$ and $n$ only. The proof is by simultaneous induction on $m+n$, and the same statement for a single element gives an $m$-bounded version. Passing to locally finite groups by an inverse-limit argument settles Hartley's problem: a locally finite group with an element whose centralizer has order $m$ has a locally soluble subgroup of finite $m$-bounded index carrying a finite normal series of $m$-bounded length with locally nilpotent factors.

What carries the argument

The proof is carried by a simultaneous induction on $m+n = |C_G(\varphi)| + |\varphi|$ proving two statements: (a) the Fitting height of $G$ is $(m,n)$-bounded, and (b) the Fitting height of any normal $\varphi$-invariant subgroup $N$ with $|C_{G/N}(\varphi)| = |C_G(\varphi)|$ is $(m,n)$-bounded. The decisive mechanism is Lemma 3.2, which shows that the equality condition forces $\varphi$-invariant Hall $\pi$-subgroups of $N$ for every prime set $\pi$, and the Busetto--Jabara inequality $h(G) \le h(G_\sigma)+h(G_\tau)+h(G_\nu)-2$ for a group factorized as $G = G_\sigma G_\tau = G_\sigma G_\nu = G_\tau G_\nu$ into three Hall subgroups. These ingredients let the induction climb the Fitting series; the base cases are Dade's theorem for fixed-point-free automorphisms (with Jabara's quadratic bound) and the biprimary case of automorphisms of order $p^a q^b$.

What would settle it

Construct a sequence of finite soluble groups $G_i$, each admitting an automorphism $\varphi_i$ of a fixed order $n$ with a fixed number $m$ of fixed points, whose Fitting heights $h(G_i)$ tend to infinity. Theorem 1.1 asserts this is impossible; for instance, any such sequence with $n=6$ and $m=2$ would be a direct counterexample.

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Extended reading notes

Core claim

The central discovery is that the invariants 'order of automorphism' and 'number of fixed points' together bound the Fitting height of a finite group, not merely the index of a soluble subgroup. Theorem 1.1 asserts that a finite group $G$ with an automorphism $\varphi$ of order $n$ and $|C_G(\varphi)| = m$ has a soluble subgroup of index and Fitting height bounded in terms of $m$ and $n$. As a corollary, any finite group with an element whose centralizer has order $m$ has such an $m$-bounded soluble subgroup, and a locally finite group with an element whose centralizer has order $m$ has a locally soluble subgroup of finite $m$-bounded index with a normal series of $m$-bounded length whose factors are locally nilpotent. This is the affirmative answer to Hartley's problem, recorded in the Kourovka Notebook.

Load-bearing premise

The proof depends, without reproving it, on the deep theorem cited as [10] that a finite group with an automorphism of order $n$ and $m$ fixed points has a soluble subgroup of index bounded in terms of $m$ and $n$; if that bound were not a true function of $m$ and $n$, the bounded-index conclusions of the corollaries would collapse.

Editorial extensions

If this is right

  • Theorem 1.1 yields a soluble subgroup in every finite group admitting an automorphism of order $n$ with $m$ fixed points, with both the index and the Fitting height bounded by functions of $m$ and $n$ alone.
  • Corollary 1.2 gives the same $m$-bounded conclusion for a finite group containing an element whose centralizer has order $m$.
  • Corollary 1.3 settles Hartley's problem: a locally finite group with an element having a finite centralizer has a locally soluble subgroup of finite $m$-bounded index with a finite normal series of $m$-bounded length and locally nilpotent factors.
  • As the paper notes, the stronger statement that a subgroup of $(m,n)$-bounded index has Fitting height bounded by $|\varphi|$ alone remains open except for automorphisms of prime-power order.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the same inverse-limit argument should give an analogous locally soluble by finite conclusion for profinite groups with an automorphism whose fixed-point subgroup is finite, provided the bounds are independent of the group.
  • Because the induction only uses the equality $|C_{G/N}(\varphi)| = |C_G(\varphi)|$, one can view the fixed-point count as a monotone invariant under quotients; this suggests that a more general 'few fixed points' theory could be built by tracking when the count drops.
  • A natural testable sharpening would be to check whether the Fitting-height bound in Theorem 1.1 can be made to depend only on the composition length $\alpha(\langle\varphi\rangle)$, as in the fixed-point-free case; the paper does not attempt this.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper proves Theorem 1.1: if a finite group G admits an automorphism φ of order n with m = |C_G(φ)| fixed points, then G contains a soluble subgroup whose index and Fitting height are bounded in terms of m and n only. The proof first invokes Hartley's classification-dependent theorem to reduce to the soluble case, and then proves a simultaneous-induction proposition (Proposition 3.1): for a soluble group G, the Fitting height of G is (m,n)-bounded, and moreover the Fitting height of any normal φ-invariant subgroup N with |C_{G/N}(φ)| = |C_G(φ)| is (m,n)-bounded. The new technical content is Lemma 3.2, which shows the hypothesis |C_{G/N}(φ)| = |C_G(φ)| forces N ⊆ I_G(φ) and hence the existence of φ-invariant Hall p'-subgroups of N, and Lemma 3.3, which bounds the Fitting height of such a Hall subgroup via Thompson's coprime theorem and the induction hypothesis; the Busetto–Jabara three-Hall-subgroup theorem then bounds h(N), and part (a) follows by slicing the Fitting series at level g(m,n). Corollary 1.2 gives the analogous statement for a single element, and Corollary 1.3 gives the affirmative answer to Hartley's problem (Kourovka Notebook Problem 13.8(a)): a locally finite group with an element whose centralizer has order m has a locally soluble subgroup of finite m-bounded index carrying a normal series of m-bounded length with locally nilpotent factors, obtained by the standard inverse-limit argument.

Significance. This is a substantial result: it settles a roughly thirty-year-old recorded problem and extends the previously known biprimary case (Hartley; Khukhro) to automorphisms of arbitrary order. The proof is clean and deliberately avoids explicit bounds, which is appropriate since only the dependence on m and n is needed for the locally finite corollary. The paper is honest about its reliance on deep external input — Hartley's generalized Brauer–Fowler theorem (CFSG-dependent), Dade's theorem, Thompson's theorem, and the Busetto–Jabara Fitting-height bound — and clearly separates the genuinely new reduction (Lemmas 3.2 and 3.3 and the Fitting-series slicing argument) from quoted results. The new tools are simple and likely to be reusable. The remarks on sharper conjectured bounds and on open problems in the nilpotent case are measured and useful. Assuming the gap identified in Major Comment 1 is repaired, this is a strong contribution to the theory of groups with few fixed points and to locally finite group theory.

major comments (1)
  1. [Section 3, proof of Proposition 3.1(b), paragraph after Lemma 3.3] The induction step of part (b) fixes "any three different primes p, q, r dividing n" and applies Theorem 2.2 to the Hall subgroups N_{p'}, N_{q'}, N_{r'}, whose corresponding subsets of π(N) are σ = π(N)\setminus{p}, τ = π(N)\setminus{q}, ν = π(N)\setminus{r}. Theorem 2.2 explicitly requires σ, τ, ν to be three different subsets of π(N), and this is not guaranteed by the hypotheses: since p, q, r divide n but need not divide |N|, one can have |π(N)| = 1 (where at most two distinct subsets exist), or |π(N)| = 2 with two of the three primes outside π(N) (so two of σ, τ, ν both equal π(N)), or more generally π(N) ∩ π(n) too small; for example, if π(N) = {2} and n = 3·5·7, then all three subsets are {2}. Nothing in the hypotheses of part (b) prevents these cases, because N is an arbitrary normal φ-invariant subgroup with |C_{G/N}(φ)| = |C_G(φ)|. Thus the Busetto–Jabara step, which is the core of the bound for h(N), is incomplete as written. The gap is localized and repairable: if some prime p dividing n does not divide |N|, then N_{p'} = N and Lemma 3.3 already yields an (m,n)-bound for h(N); otherwise π(n) ⊆ π(N), so |π(N)| ≥ 3 (since n has at least three distinct prime divisors), and then any three distinct primes dividing n give three distinct subsets, so the Busetto–Jabara argument applies as written. I recommend that the author insert this case distinction (or an equivalent argument) into the proof.
minor comments (5)
  1. [Section 3, first paragraph of the proof of Theorem 1.1] The text asserts that Hartley's theorem [10] gives a φ-invariant soluble subgroup H of (m,n)-bounded index. The theorem as quoted in the introduction provides a soluble subgroup of bounded index without φ-invariance; since invariance is used later, the proof should either quote the theorem in a form containing it or add the standard one-line argument (intersect H with its finitely many φ-conjugates and note that the index bound remains (m,n)-bounded).
  2. [Lemma 2.1(c)] The equality case in the proof of Lemma 2.1(c) would be easier to follow if written out explicitly: equality in (2.1)–(2.2) forces the full inverse image of each element of I_{G/N}(φ) to lie in I_G(φ), and applying this to the fiber over 1 = [1, φ] gives N ⊆ I_G(φ).
  3. [End of the proof of Proposition 3.1] The assertion that F_{g(m,n)+1}(G) has Fitting height greater than g(m,n) relies on the standard fact h(F_k(G)) = k for k ≤ h(G); a one-line parenthetical justification would make the argument clearer.
  4. [Theorem 2.2] The statement of Theorem 2.2 is quoted for a "finite group," but the Fitting height h(·) is defined only for finite soluble groups and Hall subgroups for arbitrary subsets of π(G) need not exist without solubility; the statement should say "finite soluble group," as in the original source [5].
  5. [Throughout] Minor typographical issues: "Hartley' theorem" should be "Hartley's theorem" in the first paragraph of Section 3; "Cambrisge" should be "Cambridge" in reference [23]; and in the abstract and Corollary 1.3 the phrase "finite m-bounded length" reads better as "m-bounded length."

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theorem is derived from external classification results, an independently corroborated biprimary base case, and well-founded induction; the only defect found is a patchable correctness gap in the Busetto–Jabara application, not a circular step.

full rationale

The derivation chain is self-contained relative to genuinely external theorems, and no step reduces the main claim to its own inputs by construction. The reduction to the soluble case invokes Hartley's generalized Brauer–Fowler theorem [10], an external result based on the classification of finite simple groups whose (m,n)-bounded index is not derived in this paper; the φ-invariance requested in the first paragraph of the proof of Theorem 1.1 is a standard one-line fix (intersect with the finitely many φ-conjugates). The base cases are external or independently established: the case m = 1 uses Dade's theorem [6] and Jabara [15], while Theorem 2.3 (the biprimary case) is the author's own result [22] but was independently proved by Hartley [11], so as a strictly weaker special case it is a genuine reduction. The simultaneous induction on m + n is well-founded: Lemma 3.3 applies the part-(a) hypothesis to the smaller instance (m', n') with |C| ≤ m and n' = n/p^k < n, and part (a) combines part (b) at (m, n) with the part-(a) hypothesis at (m − 1, n). No parameter is fitted and no quantity is normalized; the functions f(m, n) and g(m, n) arise from recursive induction. Per review policy, an omitted proof is flagged explicitly: in the proof of Proposition 3.1(b) the text states 'we pick any three different primes p, q, r dividing n' and then 'By the Busetto–Jabara Theorem 2.2', but Theorem 2.2 requires σ, τ, ν to be three different subsets of π(N), which need not hold when some of p, q, r do not divide |N| or when |π(N)| ≤ 2; the case is easily patched (if |π(N)| ≤ 1 then N is nilpotent, and if a Hall p'-subgroup equals N, Lemma 3.3 already bounds h(N)). That omitted case distinction is a correctness gap, not circularity, so it does not raise the circularity score. The non-explicitness of the bounds acknowledged in Remark 3.4 is likewise a limitation on effectiveness, not a circular step.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests entirely on a chain of published theorems, including the classification-based Hartley theorem and the author's own earlier biprimary theorem [22] (openly cited, and independently due to Hartley [11]). There are no free parameters fitted to data and no invented entities; the bounding functions f(m,n) and g(m,n) are constructed recursively inside the proof, not imported or tuned.

assumptions (8)
  • domain assumption The classification of finite simple groups is valid (used through Hartley's theorem).
    Invoked in the proof of Theorem 1.1 (Section 3, first paragraph) via Hartley's theorem [10], which gives the initial reduction to the soluble case; the paper does not re-prove this.
  • standard math Hartley's generalized Brauer-Fowler theorem [10]: a finite group with an automorphism of order n and m fixed points has a soluble subgroup of (m,n)-bounded index.
    Cited as the starting reduction in the proof of Theorem 1.1; load-bearing for the entire argument.
  • standard math Dade's theorem [6] and Jabara's bound [15]: a finite soluble group with a fixed-point-free automorphism has Fitting height bounded in terms of the number of prime factors of the automorphism order.
    Used as the base case m = 1 of the induction in Proposition 3.1, with f(1,n) = g(1,n) = 7α(⟨φ⟩)^2.
  • standard math Theorem 2.3 (Hartley [11], Khukhro [22]): the biprimary case of Theorem 1.1.
    Used in the proof of part (b) of Proposition 3.1 when n = p^a q^b; published in [22] and independently in Hartley's unpublished manuscript [11].
  • standard math Thompson's theorem [39] on coprime automorphisms of soluble groups.
    Used in Lemma 3.3 to bound h(N_p') in terms of |φ_p| and h(C_N_p'(φ_p)); coprimality holds because φ_p has order a p-power while N_p' is a Hall p'-subgroup.
  • standard math Busetto-Jabara inequality [5, Theorem 2.2] for Fitting heights of groups factorized by three Hall subgroups.
    Used in the proof of part (b) of Proposition 3.1 to bound h(N) by h(N_p') + h(N_q') + h(N_r') - 2.
  • standard math Existence and conjugacy of Hall subgroups in finite soluble groups.
    Used in Lemma 3.2 to obtain S^φ = S^a and then the φ-invariant Hall subgroup S^(b^-1).
  • standard math Finiteness of the inverse limit of finite sets of subgroups (Kegel and Wehrfritz [18, Theorem 1.K.1]).
    Used in the proof of Corollary 1.3 to pass from finite subgroups of G to a global subgroup of bounded index.

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Pith. "Pith review of On a problem of B. Hartley about a small centralizer in finite and locally finite groups." pith.science (2026). https://pith.science/paper/SIOSX55N

@misc{pith2026250520999,
  author       = {Pith},
  title        = {Pith review of: On a problem of B. Hartley about a small centralizer in finite and locally finite groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SIOSX55N}},
  note         = {Machine review of arXiv:2505.20999}
}
abstract

It is proved that if a finite group $G$ has an automorphism of order $n$ with $m$ fixed points, then $G$ has a soluble subgroup whose index and Fitting height are bounded in terms of $m$ and $n$. As a corollary, a problem of B. Hartley is solved in the affirmative: if a locally finite group $G$ has an element with finite centralizer, then $G$ has a subgroup of finite index which has a finite normal series with locally nilpotent factors.

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