When is the Sum of Two Closed Subgroups Closed in a Locally Compact Abelian Group

Francesco G. Russo; Karl H. Hofmann; Wolfgang Herfort

REVIEW 3 major objections 2 minor 14 references

When is the Sum of Two Closed Subgroups Closed in a Locally Compact Abelian Group

T0 review · 3 major / 2 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every periodic locally compact abelian group with the closed-sum property decomposes into four prime-classified pieces, discrete, profinite, finite-product, and rank-one.

desk verdict Completes Mukhin's classification for periodic LCA groups with a genuinely new four-factor theorem; one key decomposition is asserted rather than proved, but it is valid and the paper deserves review. read the letter →

arxiv 1908.08420 v1 pith:VODP7XL5 submitted 2019-08-22 math.GR

Wolfgang Herfort , Karl H. Hofmann , Francesco G. Russo This is my paper · ORCID

classification math.GR MSC 22D05

keywords locallycompactabeliangroupsclosedsubgroupsumstopologicallymodularstronglyquasihamiltonianperiodicp-groupsinductivelymonotheticMukhinclassification

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

This paper asks when a locally compact abelian group has the property that the sum $X+Y$ of any two closed subgroups is again closed. For periodic groups—totally disconnected groups that are unions of compact subgroups—it gives a complete classification: such a group splits into four pieces indexed by a partition of its primes, one discrete, one profinite, one a finite product of special $p$-groups, and one whose $p$-components each have rank one. The result extends the 1970 classification in [10] and corrects an earlier structural theorem in the authors' own book. The paper also shows that for large classes of locally compact abelian groups, topological modularity automatically forces the stronger closed-sum property.

What carries the argument

The load-bearing device is the four-way partition of the prime set $\pi(A)$ cut out by an open compact subgroup $U$: $\delta$ collects primes with $A_p\cap U=\{0\}$, $\gamma$ primes with $A_p\le U$, $\varphi$ the remaining primes of $p$-rank at least two, and $\mu$ the remaining primes of $p$-rank one. The argument reduces the whole group to these prime-component classes and then imports two obstructions: the local product $(Z(p^2),pZ(p^2))^{\mathrm{loc},\mathbb N}$ is not topologically modular, and the group $\bigoplus_I Z(p_i)\times \prod_I Z(p_i)$ is strongly topologically quasihamiltonian only for finite $I$. Those examples force the $\varphi$-class to be finite and pin down $A_\mu$ as inductively monothetic.

What would settle it

Construct a periodic locally compact abelian group $A$ that is strongly topologically quasihamiltonian and has infinite $\pi(A)$, but for which the natural map from the algebraic four-factor sum $A_\delta\oplus A_\gamma\oplus A_\varphi\oplus A_\mu$ to $A$ is not a homeomorphism—for instance, a nontrivial local product over infinitely many primes of rank-one $p$-groups. If such a group exists, the decomposition in Theorem 1.5(B) fails; if a proof shows it cannot exist, the missing topological-splitting premise is repaired.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.5: a periodic locally compact abelian group $A$ is strongly topologically quasihamiltonian—every sum of two closed subgroups is closed—if and only if its set of primes $\pi(A)$ admits a partition $\delta\cup\gamma\cup\varphi\cup\mu$ whose four direct summands have prescribed types. The pieces are a discrete group $A_\delta$, a profinite group $A_\gamma$, a finite direct sum $A_\varphi$ of strongly topologically quasihamiltonian $p$-groups each of $p$-rank at least two, and an inductively monothetic group $A_\mu$ in which every finitely generated subgroup is topologically generated by one element, so each $p$-component has $p$-rank one. In symbols, $A = A_\delta \oplus A_\gamma \oplus A_\varphi \oplus A_\mu$ topologically and algebraically. The authors present this as their genuine contribution, completing the totally disconnected periodic case of the classification in [10] and repairing Theorem 14.22(B) of their earlier book, which omitted the profinite factor $A_\gamma$.

Load-bearing premise

The classification rests on assuming that partitioning the primes of a periodic locally compact abelian group forces the group to be a topological direct product of the four prime-class pieces, a step the proof of Theorem 1.5 asserts without a separate lemma or reference.

Editorial extensions

If this is right

For locally compact abelian $p$-groups, topological modularity, the two rank conditions on an open compact subgroup, and the closed-sum property are equivalent (Theorem 1.1).
For totally disconnected non-periodic locally compact abelian groups, topologically modular implies strongly topologically quasihamiltonian, so the modular-lattice condition is no weaker than sum-closedness in that setting (Theorem 1.4).
A periodic locally compact abelian group with the closed-sum property is never a nontrivial local product over infinitely many primes: the $\varphi$-class is finite, and the remaining infinite-prime behaviour sits in discrete, profinite, or rank-one summands.
The structure theorem dualizes: under the paper's conditions, a group and its Pontryagin dual are strongly topologically quasihamiltonian together (Corollary 4.14).
The four-factor normal form supersedes the earlier description in the authors' book, adding the profinite $\gamma$-factor that was missing.

Reading between the lines

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

If the four-factor split is the whole story, the bad closed-sum behaviour in the periodic case is confined to local-product formations of infinite $p$-rank; one testable consequence is that every periodic strongly topologically quasihamiltonian group should have an open compact subgroup in which no copy of the local product $(Z(p^2),pZ(p^2))^{\mathrm{loc},\mathbb N}$ can be manufactured.
The topological splitting step asserted in the proof of Theorem 1.5 is the natural place to look for a counterexample; if a periodic locally compact abelian group with the closed-sum property and infinite prime set fails to be a topological direct product of its prime components, the classification would need an additional class of local-product factors.
Because the authors cite nonabelian locally compact groups with commuting subgroups as motivation, the abelian four-factor normal form could serve as a tool for reducing questions about such nonabelian groups to the four prime classes.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

This paper completes Mukhin's 1970 classification of locally compact abelian groups in which every sum of two closed subgroups is closed. The genuinely new piece is Theorem 1.5: periodic strongly topologically quasihamiltonian groups decompose into four pieces indexed by primes — discrete, profinite, finite-rank p-primary, and inductively monothetic — and the profinite gamma factor was missing from the authors' own 2018 book. They also correct two other results in that book. That is honest self-correction, and the p-group part of the proof is worked through in detail.

The main soft spot is the one flagged in the stress-test. In the proof of Theorem 1.5, after defining delta, gamma, and delta-prime, the paper says 'It follows that A = A_delta x A_gamma x A_delta-prime' with no lemma or reference. This is load-bearing for the classification. Having checked it against Definition 2.12, I believe the assertion is correct: a local product over a disjoint union of prime sets is the direct product of the local products over the parts, because the open compact subgroup U restricts to the product of the U intersect A_p. So the topology does split. But the paper should have stated this as a lemma; the reader is left to reconstruct a non-obvious step. It is a gap in exposition, not a mathematical error.

The converse direction of Theorem 1.2 is also only sketched, with a pointer to Mukhin's paper. The sketch is serviceable — it shows how the modular law reduces to the p-components — but again, a reader who wants the details has to go to a 1970 paper. For a paper that claims to reprove Mukhin's results, this is a bit thin.

On the positive side, the structure is clear, the notation is heavy but consistent, and the corrections to [6] are explicitly located. The reliance on the authors' own earlier work is real but not circular: the cited lemmas are published and independent of the new classification. I found no fabricated examples or fit-to-data issues; this is a structural classification, not a numerical claim.

Who benefits: researchers working on subgroup lattices of locally compact groups, especially the totally disconnected case. The paper deserves a serious referee — it will be useful and it corrects the literature. My recommendation: send it to review, but ask the referee to insist that the missing lemma about local-product splitting be added, and that the Theorem 1.2 converse be either proved or clearly delegated.

Referee Report

3 major / 2 minor

Summary. This paper studies locally compact abelian (LCA) groups in which the sum of any two closed subgroups is closed, called strongly topologically quasihamiltonian. The main theorems provide: a structural classification of p-groups (Theorem 1.1), a reduction of topological modularity of periodic groups to their p-components (Theorem 1.2), a two-partition decomposition of torsion modular groups (Theorem 1.3), a description of totally disconnected non-periodic groups (Theorem 1.4), and a four-partition classification of periodic strongly topologically quasihamiltonian groups (Theorem 1.5), together with a dual classification for groups with nontrivial connected component (Theorem 3.8). The paper aims to reprove and extend Mukhin's 1970 results and to correct errors in the authors' earlier book [6].

Significance. If correct, Theorem 1.5 completes the classification of totally disconnected LCA groups with closed sums of closed subgroups, a substantial extension of Mukhin's work. The four-factor decomposition into δ, γ, φ, and μ parts is precise and testable, and it explicitly corrects [6, Theorem 14.22(B)]. The authors supply detailed proofs for the p-group case, including the construction of local-product obstructions (Examples 2.11 and 2.13), and they are careful to identify where earlier statements err. The paper's systematic use of local products and p-rank tools is of independent interest. However, the significance depends on closing the topological splitting gap in the proof of Theorem 1.5 described below.

major comments (3)

[Proof of Theorem 1.5 (Section 4)] The assertion 'It follows that A = Aδ × Aγ × Aδ′' is the load-bearing topological step, but it is stated without proof. For infinite sets δ or δ′, the local-product topology of A (Definition 2.12) need not a priori coincide with the product of the topologies on Aδ, Aγ, and Aδ′. A lemma should be stated and proved showing that the algebraic p-primary decomposition is topological; for instance, write A as the local product of (A_p, A_p∩U), observe that U corresponds to {0}×Aγ×∏_{p∈δ′}(A_p∩U), and verify that this open subgroup determines the product topology. Without such a proof, the classification in condition (v) could miss valid groups.
[Proof of Theorem 1.5, φ-finiteness argument] The identification L/N ≅ ⊕_{p∈φ} Z(p) × ∏_{p∈φ} Z(p) is asserted without proof. The subgroup N generated by the elements pa_p must be shown to be closed, and the resulting quotient topology must be shown to make the displayed isomorphism a topological one. This is needed before Lemma 2.17 can be applied to conclude that φ is finite.
[Proof of Theorem 1.2 (Section 3.2)] The proof of the converse direction of Theorem 1.2 is only a sketch: the group G is written as a local product, but the equalities X∨Y = ⟨X_p∨Y_p⟩ and X∧Y = ⟨X_p∧Y_p⟩ are stated without proof, and the reader is referred to [10] for details. Since Theorem 1.2 is announced as one of the main results and the paper aims to reprove Mukhin's classification, a full proof or a complete 'following [10]' argument should be provided.

minor comments (2)

[Example 2.11] The line 'ι(S) = P' should read '\overline{ι(S)} = P'; as printed, the equality is false because ι(S) is a proper dense subgroup of P, not all of P.
[Theorem 3.8(b.2)] The condition 'Zφ = ∏_{p∈φ} Zφ is torsion-free' appears to contain a typo; it should likely be 'Zφ = ∏_{p∈φ} Z_p'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the main classification is derived, though one decomposition step is terse.

full rationale

Score 1: no significant circularity. The derivation chain for the main classification (Theorem 1.5) is self-contained in the relevant sense: strong topological quasihamiltonicity is not built into the structural conditions (B) by definition, and each implication is proved from local-product decompositions, subgroup/quotient heredity (Proposition 2.19), and explicitly proved auxiliary results such as Lemma 2.17. The authors' self-citations to [5], [6], [7] are to technical p-group facts (finite generation equals compact plus finite p-rank; equivalence of p-rank definitions; divisible subgroups in certain decompositions) that do not presuppose the closure-of-sums property or the theorem being proved. The most delicate line in the proof of Theorem 1.5, 'It follows that A = Aδ × Aγ × Aδ′', is a terse structural assertion rather than a circular one: it does not assume condition (B)(v) but derives a splitting from the given open compact subgroup U and the local-product convention of Definition 2.12; its brevity is a proof gap, not a reduction to inputs. Lemma 2.17 is attributed to Mukhin but is independently proved in the paper and is used only to show finiteness of φ. No fitted parameters are renamed as predictions, and no uniqueness theorem from the authors' prior work is invoked to force the classification. The score reflects reliance on several self-cited lemmas and the terse decomposition step, not actual circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants or numerical parameters appear; the paper is a purely structural classification. The central proof imports standard LCA duality and finite-rank p-group structure, and it assumes one decomposition step, the direct product splitting by prime classes in Theorem 1.5, that is asserted rather than proved.

assumptions (6)

standard math Pontryagin duality for locally compact abelian groups.
Used in Remark 2.7 (modularity of the dual) and in the proof of Theorem 3.8 to pass from G to its dual and back.
standard math Vector Splitting Theorem: a connected compact abelian group is a product of a torus factor and a compact connected group.
Invoked in Lemma 2.15(a) as [9, Theorem 7.57] to force G0 compact.
standard math Carin's finite p-rank structure: every locally compact abelian p-group of finite p-rank is Q_p^m direct sum Z(p-infinity)^n direct sum Z_p^k direct sum F.
Proposition 2.3 is quoted without proof and used in Theorems 3.1, 1.1, Corollary 3.5, and Section 4.
standard math A compact abelian torsion group is profinite of finite exponent and a finite product of compact p-groups of finite exponent.
Proposition 2.1 from [9, Corollary 8.9] underpins Lemma 2.15(c) and Proposition 2.22.
standard math Every totally disconnected locally compact abelian group has a compact open subgroup.
The letter U in Theorems 1.1, 1.4, 1.5, and in most proofs comes from this standard existence result.
domain assumption A periodic locally compact abelian group with a partition of pi(A) decomposes as the corresponding topological direct product A_delta x A_gamma x A_phi x A_mu.
Theorem 1.5 proof asserts 'It follows that A = A_delta x A_gamma x A_delta_prime' with no lemma or citation; this is the load-bearing decomposition step.

how reviews work

0 comments

Cite this review

Pith. "Pith review of When is the Sum of Two Closed Subgroups Closed in a Locally Compact Abelian Group." pith.science (2026). https://pith.science/paper/VODP7XL5

@misc{pith2026190808420,
  author       = {Pith},
  title        = {Pith review of: When is the Sum of Two Closed Subgroups Closed in a Locally Compact Abelian Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VODP7XL5}},
  note         = {Machine review of arXiv:1908.08420}
}

abstract

Locally compact abelian groups are classified in which the sum of any two closed subgroups is itself closed. This amounts to reproving and extending results by Yu.~N.~Mukhin from 1970. Namely we contribute a complete classification of all totally disconnected \lca\ groups with $X+Y$ closed for any closed subgroups $X$ and $Y$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[6]

Herfort, K

W. Herfort, K. H. Hofmann, and F. G. Russo. Locally Compact Periodic Groups. A Study of a Class of Totally Disconnected Topologic al Groups. Num- ber 71 in Studies in Mathematics. Berlin, Boston: De Gruyter, 2018

work page 2018
[10]

J. N. Mukhin. Topological abelian groups with Dedekind lattice of c losed sub- groups. Mat. Zametki , 8:509–519, 1970

work page 1970
[1]

D. L. Armacost. The structure of locally compact abelian groups , volume 68 of Monographs and Textbooks in Pure and Applied Mathematics . Marcel Dekker, Inc., New York, 1981. 38 WOLFGANG HERFORT, KARL H. HOFMANN, AND FRANCESCO G. RUSSO

work page 1981
[2]

V. S. ˇCarin. On groups of ﬁnite rank. II. Ukrain. Mat. ˇZ., 18(3):85–96, 1966

work page 1966
[3]

Dedekind

R. Dedekind. ¨Uber die Anzahl der Ideal-classen in den verschiedenen Ordn un- gen eines endlichen K¨ orpers. Festschrift zur Saecularfei er des Geburtstages von C. F. Gauss. Vieweg, Braunschweig, 1877

work page
[4]

L. Fuchs. Inﬁnite abelian groups. Vol. I . Pure and Applied Mathematics, Vol

work page
[5]

Herfort, K

W. Herfort, K. H. Hofmann, and F. G. Russo. Locally Compact Gr oups with Commuting Subgroups. manuscript, 2018

work page 2018
[7]

Herfort, K

W. Herfort, K. H. Hofmann, and F. G. Russo. Locally Compact Ab elian p- Groups. Topology and its Applications , 259:203 – 241, 2019

work page 2019

Show all 14 references

[8]

Hewitt and K

E. Hewitt and K. A. Ross. Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representa tions. Die Grundlehren der mathematischen Wissenschaften, Bd. 115. Academic Press, I nc., Publish- ers, New York; Springer-Verlag, Berlin-G¨ otti...

1963
[9]

K. H. Hofmann and S. A. Morris. The structure of compact groups , volume 25 of De Gruyter Studies in Mathematics . De Gruyter, Berlin, 2013. A primer for the student—a handbook for the expert, Third edition, revised an d augmented

2013
[11]

Y. N. Mukhin. Topological groups with a Dedekind lattice of closed subgroups. Mat. Issled. , 105(105, Moduli, Algebry, Topol.):129–141, 197, 1988

1988
[12]

Ribes and P

L. Ribes and P. Zalesskii. Proﬁnite groups, volume 40 of Ergebnisse der Mathe- matik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Su rveys in Mathe- matics [Results in Mathematics and Related Areas. 3rd Serie s. A Series of Mod- ern Surveys in Mathematics] . Springer-V...

2010
[13]

R. Schmidt. Subgroup lattices of groups , volume 14 of de Gruyter Expositions in Mathematics . Walter de Gruyter and Co., Berlin, 1994. Technische Universit ¨at Wien, Wiednerhauptstraße 8-10/E101, Aus- tria, Email: wolfgang.herfort@tuwien.ac.at F achbereich Mathematik, Technis...

1994
[36]

Academic Press, New York-London, 1970

1970

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

Herfort, K

W. Herfort, K. H. Hofmann, and F. G. Russo. Locally Compact Periodic Groups. A Study of a Class of Totally Disconnected Topologic al Groups. Num- ber 71 in Studies in Mathematics. Berlin, Boston: De Gruyter, 2018

work page 2018

[2] [10]

J. N. Mukhin. Topological abelian groups with Dedekind lattice of c losed sub- groups. Mat. Zametki , 8:509–519, 1970

work page 1970

[3] [1]

D. L. Armacost. The structure of locally compact abelian groups , volume 68 of Monographs and Textbooks in Pure and Applied Mathematics . Marcel Dekker, Inc., New York, 1981. 38 WOLFGANG HERFORT, KARL H. HOFMANN, AND FRANCESCO G. RUSSO

work page 1981

[4] [2]

V. S. ˇCarin. On groups of ﬁnite rank. II. Ukrain. Mat. ˇZ., 18(3):85–96, 1966

work page 1966

[5] [3]

Dedekind

R. Dedekind. ¨Uber die Anzahl der Ideal-classen in den verschiedenen Ordn un- gen eines endlichen K¨ orpers. Festschrift zur Saecularfei er des Geburtstages von C. F. Gauss. Vieweg, Braunschweig, 1877

work page

[6] [4]

L. Fuchs. Inﬁnite abelian groups. Vol. I . Pure and Applied Mathematics, Vol

work page

[7] [5]

Herfort, K

W. Herfort, K. H. Hofmann, and F. G. Russo. Locally Compact Gr oups with Commuting Subgroups. manuscript, 2018

work page 2018

[8] [7]

Herfort, K

W. Herfort, K. H. Hofmann, and F. G. Russo. Locally Compact Ab elian p- Groups. Topology and its Applications , 259:203 – 241, 2019

work page 2019