Hopficity of profinite completions of abelian groups

E. Ingrosso; M. Brescia; M. Trombetti

Hopficity of profinite completions of abelian groups

T0 review · 0 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For abelian groups, the profinite completion is topologically Hopfian exactly when A/pA is finite for every prime p, and this criterion refutes a long-standing open problem.

desk verdict A clean, correct characterization that answers a 1978 problem; the proof is solid, and only minor presentation nits remain. read the letter →

arxiv 2607.19193 v1 pith:ZS6WAI2I submitted 2026-07-21 math.GR

M. Brescia , E. Ingrosso , M. Trombetti This is my paper · ORCID

classification math.GR MSC 20E1820K2022C05

keywords Hopfiangroupprofinitecompletionabelianpro-pp-basicsubgrouptopologicallyKourovkaProblem6.30residuallyfinite

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 pins down exactly when the profinite completion of an abelian group is topologically Hopfian: precisely when the quotient A/pA is finite for every prime p, equivalently when every Sylow pro-p subgroup of the completion is topologically finitely generated. This reduces a question about a potentially enormous completion to a check on finite quotients of the original group. The proof works prime by prime via p-basic subgroups, showing that an infinite ladder of summands in a p-basic subgroup yields a surjective non-injective endomorphism of the p-adic completion. As a byproduct, the criterion answers Kourovka Problem 6.30 in the negative: a residually finite Hopfian group can have a profinite completion that is not topologically Hopfian.

What carries the argument

The load-bearing object is the p-basic subgroup B of A (Kulikov's theorem): B is a direct sum of infinite cyclic groups and finite cyclic p-groups, p-pure in A, with p-divisible quotient A/B. Lemma 2.5 shows that restriction to B is a bijection on homomorphisms into finite p-groups, so the pro-p completions of A and B are isomorphic and A/pA ≅ B/pB. In the infinite-rank case, Proposition 3.1 constructs a surjective non-injective endomorphism of the pro-p completion by assigning each summand a 'height' (∞ for infinite cyclic, n for C_{p^n}), picking an infinite non-decreasing ladder, shifting each selected summand to the next, and letting the universal property of pro-p completion extend the

What would settle it

A single abelian group with A/pA finite for every p but with a surjective non-injective continuous endomorphism of its profinite completion would refute the theorem, as would a group with A/pA infinite for some p whose pro-p completion is nonetheless Hopfian. In the specific example A = ⊕_{i≥1} Z[1/q_i] with distinct odd primes, A/2A is infinite; the theorem predicts a non-Hopfian 2-adic completion, so either constructing or ruling out such an endomorphism would settle the prediction.

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Extended reading notes

Core claim

The central claim is a complete classification: for an abelian group A, the profinite completion Â is topologically Hopfian if and only if A/pA is finite for every prime p. Because Â decomposes as the direct product of its Sylow pro-p subgroups, the proof is prime-by-prime: Â_p is Hopfian exactly when A/pA is finite, equivalently when Â_p is topologically finitely generated. The transfer device is a p-basic subgroup B of A, which shares both the pro-p completion and the mod-p reduction with A. If B has infinitely many cyclic summands, an endomorphism of the pro-p completion is built by shifting indices of non-decreasing p-heights and killing one summand, producing a surjective but non-inject

Load-bearing premise

Everything depends on Kulikov's theorem guaranteeing that every abelian group has a p-basic subgroup and on the extension lemma that every homomorphism from that subgroup to a finite p-group lifts to the whole group; if either fails, the identification of the pro-p completions of A and B breaks and the criterion may not transfer.

Editorial extensions

If this is right

Topological Hopficity of the profinite completion of an abelian group is equivalent to topological finite generation of each Sylow pro-p subgroup.
The Kourovka Problem 6.30 answer is negative: Hopfian and residually finite do not imply the profinite completion is Hopfian; the explicit group is a direct sum of localizations Z[1/q_i] with distinct odd primes.
For abelian p-groups, a Hopfian profinite completion forces the completion to be finite and the group to split as D ⊕ F with F finite (Corollary 3.3).
For direct sums of localizations A = ⊕_{i} Z[S_i^{-1}], Hopficity of the completion is equivalent to finiteness, for each prime p, of the set of indices i with p not in S_i (Corollary 3.4).
In the abelian setting, non-Hopficity of the completion is detected already by an infinite elementary abelian p-quotient A/pA.

Reading between the lines

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

The same local criterion may serve as a sufficient condition for Hopficity of profinite completions in larger classes, such as finitely generated nilpotent or polycyclic groups, where the mod-p abelianization could play the role of A/pA; the abelian proof suggests the completion is non-Hopfian whenever the mod-p abelianization is infinite.
The counterexample shows that 'residually finite plus Hopfian' is not a robust combination under profinite completion; a more natural sufficient condition for a Hopfian completion is that each pro-p Sylow subgroup is topologically finitely generated, exactly the third condition of the Main Theorem.
The height-shift construction is a general recipe for producing non-injective surjective endomorphisms in completions of infinite-rank abelian groups; a similar ladder argument might yield analogous examples in nonabelian pro-p groups with an infinite descending chain of open subgroups, though the abelian structure is used crucially.
Because the criterion is purely in terms of finite quotients A/pA, it is effectively computable for finitely presented abelian groups via Smith normal form; this suggests treating Hopficity of profinite completions as a decidable property in that class.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the main theorem is true and the paper earns its accept. The characterization of Hopficity of profinite completions of abelian groups in terms of finiteness of A/pA is new and answers a 1978 Kourovka problem in the negative. The proof is largely built on Kulikov's p-basic subgroup theorem, which is standard and applied correctly. The genuinely original mechanism is Proposition 3.1, the shift endomorphism on an infinite p-basic direct sum; it is the right construction and the continuity argument checks out.

The paper does well by spelling out Lemma 2.5 in detail: the extension of homomorphisms from a p-basic subgroup to finite p-groups is carefully justified, and the isomorphism of pro-p completions follows properly. The density/closed-image argument is sound. Giving the equivalence with all Sylow pro-p subgroups topologically finitely generated is a nice bonus.

Soft spots are minor. The counterexample G = ⊕ Z[1/q_i] is said to be "clearly" Hopfian and residually finite. It is true, but a one-sentence justification would make the paper more self-contained. Also, the Hopfian-implies-finiteness direction depends entirely on Proposition 3.1; that is fine, but the paper could acknowledge this is the only non-structural input. The reliance on Kulikov's theorem is significant but standard, and the authors use it exactly as intended.

I found no load-bearing flaw. The stress test checked the extension lemma and the shift construction; both hold. The citation pattern is honest: no self-citation, no fitting. The external dependencies are classical and correctly applied. If the referee verifies Lemma 2.5 (which is proved) and the application of Kulikov, the theorem is solid.

Who is this for? Any group theorist working with Hopfian groups, profinite completions, or abelian groups. It is a clean result that will be a useful reference. I would not be surprised if the proof can be streamlined, but it is already concise.

Recommendation: send it to a serious referee. The result is significant enough and the proof looks correct. I would accept with minor revisions, mostly the one-sentence justification of the example.

Referee Report

0 major / 6 minor

Summary. The paper gives an exact criterion for the profinite completion of an abelian group to be topologically Hopfian. Main Theorem: $\widehat A$ is topologically Hopfian iff $A/pA$ is finite for every prime $p$ iff every Sylow pro-$p$ subgroup of $\widehat A$ is topologically finitely generated. The proof passes through p-basic subgroups: after reducing to a p-basic subgroup $B$, the quotient $A/pA$ controls finite generation of $B$; if $B$ is infinite, the authors construct an explicit shift endomorphism of $\widehat B_p$ that is surjective but not injective. The paper also gives a counterexample to Kourovka Problem 6.30: for distinct odd primes $q_i$, the group $G=\bigoplus_{i\ge 1}\mathbb Z[1/q_i]$ is residually finite and Hopfian, but $\widehat G$ is not topologically Hopfian because $G/2G$ is infinite.

Significance. The characterization is complete and checkable, and the counterexample resolves a long-standing question of Mel'nikov. The proof is self-contained modulo standard theorems (Kulikov's p-basic subgroup theorem, Hopficity of finitely generated profinite groups). No parameters are fitted, no external results are assumed beyond standard ones, and the arguments in Lemma 2.5 and Proposition 3.1 are detailed and correct. The shift construction for infinite p-basic subgroups is elegant and directly yields the non-Hopfian completion. This is a valuable contribution to the theory of profinite completions and Hopfian groups.

minor comments (6)

[Introduction, counterexample] The assertions that $G=\bigoplus_i \mathbb Z[1/q_i]$ is 'clearly Hopfian and residually finite' are true, but a short justification would be helpful. In particular, for $i\neq j$ there are no nonzero homomorphisms $\mathbb Z[1/q_i]\to \mathbb Z[1/q_j]$, so every endomorphism preserves the direct summands; each summand is Hopfian. A one-sentence explanation would also clarify why this example is not vacuous (compare with $\bigoplus_{i\ge1}\mathbb Z$, which is not Hopfian).
[Theorem 3.2] The notation 'Zrp⊕Fp' is misleading: it should be $\mathbb Z^r \oplus F_p$, not $\mathbb Z_p^r \oplus F_p$. Please correct the typesetting.
[Lemma 2.2] The proof of Lemma 2.2 is omitted. Since this lemma is used in the proof of the Main Theorem, a two-line argument (continuous endomorphisms preserve the Sylow pro-$p$ subgroups, and surjectivity on the product is equivalent to surjectivity on each factor) would make the paper more self-contained.
[Proposition 3.1] The sentence 'Clearly, $\sigma$ is continuous with respect to the pro-$p$ topology on $B$' is correct, but a brief parenthetical explanation (the preimage of an open subgroup of $p$-power index has $p$-power index, hence is open) would help the reader.
[Lemma 2.5(ii)] In the injectivity argument, the statement 'some finite $p$-quotient of $B$ does not annihilate $x$' relies on the fact that the kernels of all finite $p$-quotients have trivial intersection. This standard fact could be stated explicitly.
[References] Theorem 2.4 is cited to 'the chapter on purity and basic subgroups in [1]'; a more precise theorem number or page reference would be convenient.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained from standard external theorems and does not reduce to its own inputs.

full rationale

The paper's central claim is that Hopficity of the profinite completion of an abelian group A is equivalent to finiteness of A/pA for every prime p. The proof proceeds by reducing to Sylow pro-p subgroups, then to p-basic subgroups via Kulikov's theorem (Theorem 2.4). Lemma 2.5, whose proof is included, establishes the isomorphism \widehat B_p ≅ \widehat A_p for a p-basic subgroup B, using only the definition of p-basic subgroup and elementary properties of p-divisible groups; this is not circular because the bijection Hom(A,F)≅Hom(B,F) is proved from the defining purity/divisibility conditions, not assumed from the target theorem. Proposition 3.1 independently constructs a non-injective surjective endomorphism of \widehat B_p when B is not finitely generated, and the converse uses the standard fact that topologically finitely generated profinite groups are Hopfian. No parameter is fitted and later called a prediction; no ansatz is imported from the authors' own prior work; no self-citations are load-bearing, since the only references are standard external texts (Fuchs, Ribes–Zalesskii, Robinson, Kourovka Notebook). The counterexample asserting that ⊕ ℤ[1/q_i] is Hopfian and residually finite is stated without proof, but that claim is not an input to the main derivation and the non-Hopficity of its completion follows from the independently proved criterion. Thus the derivation chain does not reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities appear. The derivation rests on standard structural theorems of abelian groups and profinite groups, all of which are external to the paper's central claim.

assumptions (4)

standard math Every abelian group has a p-basic subgroup for every prime p (Kulikov's theorem, Theorem 2.4).
Used in Theorem 3.2 to reduce ĤA_p to the completion of a p-basic subgroup B; without this structure theorem the central reduction collapses.
standard math Every topologically finitely generated profinite group is topologically Hopfian (Lemma 2.1).
Used to prove that finite generation of each ĤA_p implies Hopficity; stated without proof as a known fact.
standard math The profinite completion of an abelian group splits as ĤA ≅ ∏_p ĤA_p, with ĤA_p the Sylow pro-p subgroup.
Standard profinite-group background from Ribes–Zalesskii; used to reduce Hopficity prime by prime.
standard math B is residually a finite p-group, so B embeds canonically into ĤB_p.
Used in Proposition 3.1 to show the constructed endomorphism of ĤB_p has nontrivial kernel; follows from B being a direct sum of cyclic groups.

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Cite this review

Pith. "Pith review of Hopficity of profinite completions of abelian groups." pith.science (2026). https://pith.science/paper/ZS6WAI2I

@misc{pith2026260719193,
  author       = {Pith},
  title        = {Pith review of: Hopficity of profinite completions of abelian groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZS6WAI2I}},
  note         = {Machine review of arXiv:2607.19193}
}

abstract

We determine exactly when the profinite completion of an arbitrary abelian group is topologically Hopfian. For an abelian group $A$, we prove that \[ \widehat A \text{ is topologically Hopfian} \quad\Longleftrightarrow\quad A/pA \text{ is finite for every prime }p. \] As a byproduct, we answer Problem 6.30 of the Kourovka Notebook in the negative: for pairwise distinct odd primes $q_i$, the group $\bigoplus_{i\geq1}\Z[1/q_i]$ is residually finite and Hopfian, whereas its profinite completion is not topologically Hopfian.

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Reference graph

Works this paper leans on

4 extracted references · 1 linked inside Pith

[1]

Fuchs,Abelian Groups, Springer Monographs in Mathematics, Springer International Publishing, Cham, 2015

L. Fuchs,Abelian Groups, Springer Monographs in Mathematics, Springer International Publishing, Cham, 2015

2015
[2]

E. I. Khukhro and V. D. Mazurov (eds.),Unsolved Problems in Group Theory: The Kourovka Notebook, No. 21, Sobolev Institute of Mathematics, Novosibirsk, 2026; arXiv:1401.0300

arXiv 2026
[3]

Ribes and P

L. Ribes and P. Zalesskii,Profinite Groups, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 40, Springer, Berlin, 2010

2010
[4]

Renato Caccioppoli

D. J. S. Robinson,A Course in the Theory of Groups, second edition, Graduate Texts in Mathematics, vol. 80, Springer, New York, 1996. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, Napoli, Italy Email address:mattia.brescia@unina.it Email ad...

1996

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Fuchs,Abelian Groups, Springer Monographs in Mathematics, Springer International Publishing, Cham, 2015

L. Fuchs,Abelian Groups, Springer Monographs in Mathematics, Springer International Publishing, Cham, 2015

2015

[2] [2]

E. I. Khukhro and V. D. Mazurov (eds.),Unsolved Problems in Group Theory: The Kourovka Notebook, No. 21, Sobolev Institute of Mathematics, Novosibirsk, 2026; arXiv:1401.0300

arXiv 2026

[3] [3]

Ribes and P

L. Ribes and P. Zalesskii,Profinite Groups, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 40, Springer, Berlin, 2010

2010

[4] [4]

Renato Caccioppoli

D. J. S. Robinson,A Course in the Theory of Groups, second edition, Graduate Texts in Mathematics, vol. 80, Springer, New York, 1996. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, Napoli, Italy Email address:mattia.brescia@unina.it Email ad...

1996