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arxiv: 2605.17495 · v1 · pith:5SBIEKMCnew · submitted 2026-05-17 · 🧮 math.GN · math.GR

Maximally almost periodic subgroups of Abelian groups of prime exponent

Ol'ga Sipacheva This is my paper

Pith reviewed 2026-05-19 22:34 UTC · model grok-4.3

classification 🧮 math.GN math.GR
keywords Abelian topological groupsprime exponentmaximally almost periodic subgroupstopological groupsalmost periodic propertiessubgroups
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The pith

Any infinite Abelian topological group of prime exponent contains an infinite maximally almost periodic subgroup.

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

The paper shows that every infinite Abelian topological group in which every element satisfies x to the power of a fixed prime equals the identity must contain an infinite subgroup where continuous characters to the circle group separate points. A sympathetic reader cares because this guarantees a substantial almost periodic component even when the full group may lack enough continuous characters to be maximally almost periodic itself. The result therefore supplies a uniform way to locate large subgroups amenable to representation-theoretic or duality methods within the class of bounded-exponent groups. It applies precisely when the group is topological, Abelian, infinite, and of prime exponent.

Core claim

It is proved that any infinite Abelian topological group of prime exponent has an infinite maximally almost periodic subgroup.

What carries the argument

The maximally almost periodic subgroup, a subgroup on which the continuous characters separate points.

If this is right

  • Every such group admits at least one infinite subgroup on which continuous characters separate points.
  • The almost periodic part of these groups is always infinite.
  • The result applies uniformly across all topologies that make the group Abelian of prime exponent.

Where Pith is reading between the lines

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

  • The same existence statement may extend to groups of bounded but not necessarily prime exponent under suitable extra conditions.
  • The construction supplies a concrete way to produce non-trivial continuous characters on infinite subsets, which could be used to test duality properties in concrete examples.

Load-bearing premise

The group must be infinite, Abelian, topological, and of prime exponent; without any one of these the existence need not hold.

What would settle it

An explicit construction of an infinite Abelian topological group of prime exponent whose every maximally almost periodic subgroup is finite would refute the claim.

read the original abstract

It is proved that any infinite Abelian topological group of prime exponent has an infinite maximally almost periodic subgroup.

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

1 major / 1 minor

Summary. The manuscript claims to prove that any infinite Abelian topological group of prime exponent has an infinite maximally almost periodic subgroup.

Significance. If the central claim holds after appropriate clarification of the topological hypotheses, the result would supply a structural existence theorem for MAP subgroups in a broad class of topological Abelian groups, potentially useful for representation theory and almost-periodic functions on groups of exponent p. The paper supplies no machine-checked proofs or explicit parameter-free constructions in the abstract.

major comments (1)
  1. [Abstract] Abstract: the stated theorem asserts the existence result for every infinite Abelian topological group of prime exponent. No Hausdorff (or T0/T1) separation axiom is mentioned. The indiscrete topology on any infinite elementary Abelian p-group is a valid topological-group structure; every subgroup inherits the indiscrete topology, continuous characters cannot separate points, and therefore no infinite subgroup is MAP. This counterexample shows the claim as written is false and is load-bearing for the central existence statement.
minor comments (1)
  1. The abstract supplies no lemmas, derivation steps, or verification details, which prevents direct checking of the argument even after the separation-axiom issue is addressed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and for identifying the critical omission of separation axioms in the statement of our main result. We agree that the theorem as currently phrased is incorrect without the Hausdorff assumption and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the stated theorem asserts the existence result for every infinite Abelian topological group of prime exponent. No Hausdorff (or T0/T1) separation axiom is mentioned. The indiscrete topology on any infinite elementary Abelian p-group is a valid topological-group structure; every subgroup inherits the indiscrete topology, continuous characters cannot separate points, and therefore no infinite subgroup is MAP. This counterexample shows the claim as written is false and is load-bearing for the central existence statement.

    Authors: We fully agree with this observation. In the indiscrete topology the only continuous character is the trivial homomorphism, so no infinite subgroup can be maximally almost periodic. Our proof relies on the existence of sufficiently many continuous characters, which presupposes that the group is at least Hausdorff (in fact, the argument uses that points can be separated from the identity by continuous characters). We will add the explicit hypothesis that all topological groups under consideration are Hausdorff both in the abstract and in the statement of the main theorem. This is a clarification rather than a change of content; the remainder of the argument is unaffected. revision: yes

Circularity Check

0 steps flagged

Direct existence proof in pure mathematics exhibits no circularity

full rationale

The paper claims to prove an existence result for infinite Abelian topological groups of prime exponent possessing infinite maximally almost periodic subgroups. This is a standard first-principles derivation in topological group theory that invokes external definitions of topological groups, characters, and almost periodicity without introducing fitted parameters, self-definitional loops, or load-bearing self-citations that reduce the central claim to the paper's own inputs. The derivation chain remains independent of any internal fitting or renaming of known results, making the result self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definitions of Abelian topological groups, prime exponent, and maximally almost periodic subgroups; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms and definitions of topological groups and groups of prime exponent
    Invoked implicitly by the statement that the group is Abelian, topological, and of prime exponent.

pith-pipeline@v0.9.0 · 5520 in / 1121 out tokens · 37865 ms · 2026-05-19T22:34:58.009363+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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uses
The paper appears to rely on the theorem as machinery.
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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    work page 1979

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    Arhangel’skii and M

    A. Arhangel’skii and M. Tkachenko,Topological Groups and Related Structures, (At- lantis Press, Amsterdam, 2008)

    work page 2008

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    The theory of topological groups I,

    M. I. Graev, “The theory of topological groups I,” Usp. Mat. Nauk,5(2), 3–56 (1950)

    work page 1950

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    Engelking,General Topology, 2nd ed

    R. Engelking,General Topology, 2nd ed. (Heldermann-Verlag, Berlin, 1989)

    work page 1989

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    Fuchs,Infinite Abelian Groups(Academic Press, New York, 1970), Vol

    L. Fuchs,Infinite Abelian Groups(Academic Press, New York, 1970), Vol. 1

    work page 1970

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    Direct sums and products in topological groups and vector spaces,

    D. Dikranjan, D. Shakhmatov, and J. Spˇ ev´ ak, “Direct sums and products in topological groups and vector spaces,” J. Math. Anal. Appl.437, 1257–1282 (2016)

    work page 2016

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    Minimally almost periodic groups,

    J. von Neumann and E. P. Wigner, “Minimally almost periodic groups,” Ann. Math. 41(4), 746–750 (1940)

    work page 1940

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    Topologies on abelian groups,

    E. G. Zelenyuk and I. V. Protasov, “Topologies on abelian groups,” Math. USSR-Izv. 37(2), 445–460 (1991). Email address:osipa@gmail.com Department of General Topology and Geometry, F aculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 199991 Russia

    work page 1991