On verbally closed subgroups of free solvable groups

E.I. Timoshenko; V.A. Roman'kov

arxiv: 1906.11689 · v1 · pith:FZ5JXB4Wnew · submitted 2019-06-26 · 🧮 math.GR

On verbally closed subgroups of free solvable groups

V.A. Roman'kov , E.I. Timoshenko This is my paper

Pith reviewed 2026-05-25 15:28 UTC · model grok-4.3

classification 🧮 math.GR

keywords verbally closed subgroupsfree solvable groupsretractsalgebraically closed subgroupssolvable groupsgroup varieties

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The pith

Sufficient conditions turn verbally closed subgroups of free solvable groups into retracts and thus algebraically closed subgroups.

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

The paper studies verbally closed subgroups inside free solvable groups. It proves several results supplying sufficient conditions under which such a subgroup becomes a retract of the larger group. When the retract property holds, the subgroup is algebraically closed in the full group. A reader would care because the work connects verbal closure directly to algebraic closure via retracts within the variety of solvable groups of finite derived length.

Core claim

The authors prove a number of results that give sufficient conditions under which a verbally closed subgroup is turned to be a retract and so algebraically closed of the full group.

What carries the argument

The retract property that converts a verbally closed subgroup into an algebraically closed one inside a free solvable group.

Load-bearing premise

The paper takes the standard definitions of verbally closed subgroup, retract, and free solvable group as given without re-deriving them.

What would settle it

An explicit example of a verbally closed subgroup of a free solvable group that satisfies one of the paper's sufficient conditions yet fails to be a retract would disprove the corresponding result.

read the original abstract

We study verbally closed subgroups of free solvable groups. A number of results is proved that give sufficient conditions under whose a verbally closed subgroup is turned to be a retract and so algebraically closed of the full group.

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.

Desk Editor's Note private letter to a colleague

This paper states some sufficient conditions for verbally closed subgroups of free solvable groups to become retracts, but the contribution stays narrow and incremental.

read the letter

The main result here is a set of sufficient conditions under which a verbally closed subgroup of a free solvable group of finite derived length is also a retract, hence algebraically closed in the variety. That is the concrete new piece: the authors supply explicit criteria that turn verbal closure into the stronger retract property. The work stays inside standard definitions of verbal subgroups, retracts, and free solvable groups, so it does not introduce new machinery or resolve open questions in broader group theory. What it does is tighten the relationship between these notions for this specific class, which is useful for people already working on verbal and algebraic closure in solvable groups. The proofs appear to follow from direct applications of existing facts about the variety rather than from heavy new arguments. That keeps the paper short and focused, but it also means the advance is modest. No examples are mentioned in the abstract, and without them it is difficult to see how often the conditions actually apply or whether they capture the interesting cases. The citation pattern is not visible, but the topic sits in a small corner of combinatorial group theory, so overlap with earlier work on retracts in solvable groups would need checking. Overall the claims look internally consistent and the assumptions are the usual ones. This is the sort of paper that belongs in a specialized journal like the Journal of Algebra or Algebra Colloquium rather than a general venue. A serious editor could send it to referees who know the literature on verbal subgroups; the results are narrow enough that revision would likely be light if the proofs check out. I would not bring it to a general reading group, and I would not cite it unless I were writing directly on this exact question.

Referee Report

0 major / 2 minor

Summary. The manuscript studies verbally closed subgroups of free solvable groups and proves a collection of results supplying sufficient conditions under which such a subgroup is a retract (hence algebraically closed in the ambient variety of solvable groups of finite derived length). The arguments rely exclusively on the standard definitions of verbal closure, retracts, and free solvable groups.

Significance. If the stated theorems hold, the results would add concrete sufficient conditions to the literature on algebraic closure and retracts within solvable varieties, extending known facts about free groups without introducing nonstandard assumptions or ad-hoc parameters.

minor comments (2)

Abstract: the phrasing 'under whose a verbally closed subgroup is turned to be a retract' is grammatically incorrect and should be revised for clarity (e.g., 'under which a verbally closed subgroup becomes a retract').
Abstract: the sentence structure is awkward and could be improved to better convey the main results; consider expanding slightly to indicate the nature of the sufficient conditions without revealing proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and for recommending minor revision. No specific major comments are listed in the report, so there are no individual points requiring direct response or rebuttal.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard definitions only

full rationale

The paper consists of theorems providing sufficient conditions for verbally closed subgroups of free solvable groups to be retracts (hence algebraically closed in the variety). It invokes only the standard definitions of verbal closure, retracts, and free solvable groups of finite derived length without re-derivation, fitted parameters, predictions, or load-bearing self-citations. No equations reduce to inputs by construction, and the central claims remain independent of any internal circular chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all background notions are presumed standard in group theory.

pith-pipeline@v0.9.0 · 5544 in / 869 out tokens · 15849 ms · 2026-05-25T15:28:45.542330+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

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A. Myasnikov, V. Roman ’kov , Verbally closed subgroups of free groups, J. Group Theory, 17, No. 1 (2014), 29–40

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Baumslag, A

G. Baumslag, A. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, Contemporary Math., 296 (2002), 1–38

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Retracts of free groups and a question of Bergman

I. Snopce, S. Tanushevski, P. Zalesskii , Retracts of free groups and a ques- tion of Bergman, arXiv:1902.02378 [math.GR]. Feb 6 (2019). 16

work page internal anchor Pith review Pith/arXiv arXiv 1902
[8]

Roman ’kov, N.G

V.A. Roman ’kov, N.G. Khisamiev, Verbally and existentially closed sub- groups of free nilpotent groups. Algebra and Logic. 2013. 52, No. 4, 336–351

work page 2013
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A. M. Mazhuga , On free decompositions of verbally closed subgroups in fre e products of ﬁnite groups, J. Group Theory, 20, No. 5 (2017), 971–986

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A.M. Mazhuga , Strongly verbally closed groups, J. Algebra, 493 (2018), 171–184

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Mazhuga , Verbally closed subgroups,Thesis, Lomonosov Moscow State University, Moscow, 2018 (In Russian)

A.M. Mazhuga , Verbally closed subgroups,Thesis, Lomonosov Moscow State University, Moscow, 2018 (In Russian)

work page 2018
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Ant. A. Klyachko, A.M. Mazhuga , Verbally closed virtually free subgroups, Sbornik: Mathematics, 209, No. 6 (2018), 850–856

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A.Klyachko, A

A. A.Klyachko, A. M. Mazhuga, V. Y. Miroshnichenko , Virtually free ﬁnite-normal-subgroup-free groups are strongly verbally closed, J. Algebra, 510 (2018), 319–330

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Timoshenko , Test elements and test rank of a free metabelian group, Siberian Mathematical Journal, 47, No

E.I. Timoshenko , Test elements and test rank of a free metabelian group, Siberian Mathematical Journal, 47, No. 6 (2000), 1200–1204

work page 2000
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Roman ’kov, Test elements for free solvable groups of rank 2

V.A. Roman ’kov, Test elements for free solvable groups of rank 2. Algebra and Logic. 2001. 40, No.2. 106–111

work page 2001
[16]

Timoshenko , Computing test rank for a free solvable group

E.I. Timoshenko , Computing test rank for a free solvable group. Algebra and Logic, 45,No. 4 (2006), 254–260

work page 2006
[17]

Gupta, E.I

C.K. Gupta, E.I. Timoshenko, Test rank for fome free polynilpotent Groups, Algebra and Logic 42, No. 1 (2003), 20–28

work page 2003
[18]

Timoshenko , Endomorphisms and universal theories of solvable groups

E.I. Timoshenko , Endomorphisms and universal theories of solvable groups. Novosibirsk State Technical University, Novosibi rsk, 2011 (In Rus- sian)

work page 2011
[19]

Remeslennikov, V.A

V.N. Remeslennikov, V.A. Sokolov , Some properties of Magnus embedding, Algebra and Logic, 9 (1970), 342–349

work page 1970
[20]

Crowell, R.H

R.H. Crowell, R.H. Fox , Introduction to knot theory, Springer-Verlag, Berlin-Heidelberg- New York, 1963

work page 1963
[21]

V. A. Roman ’kov, Essays in algebra and cryptology. Solvable groups, Omsk, Omsk State University Publishing House, 2017. 17

work page 2017
[22]

Baumslag , Some subgroup theorems for free υ-groups, Trans

G. Baumslag , Some subgroup theorems for free υ-groups, Trans. Amer. Math. Soc., 108 (1963), 516–525

work page 1963
[23]

Roman ’kov, Equations in free metabelian groups, Siberian Mathe- matical Journal

V.A. Roman ’kov, Equations in free metabelian groups, Siberian Mathe- matical Journal. 1979. 20, No. 3 (1979). 469–471

work page 1979
[24]

Neumann, Varieties of groups, Springer-Verlag, Berlin-Heidelber g, 1967

H. Neumann, Varieties of groups, Springer-Verlag, Berlin-Heidelber g, 1967

work page 1967
[25]

Artamonov , Projective metabelian groups and Lie algebras, Uspekhi Mat

V.A. Artamonov , Projective metabelian groups and Lie algebras, Uspekhi Mat. Nauk, 32, No. 3(195) (1977), 166

work page 1977
[26]

Lewin , A note on zero divisors in group rings, Proc

J. Lewin , A note on zero divisors in group rings, Proc. Amer. Math. Soc ., 31 (1972), 357–359

work page 1972
[27]

C. K. Gupta, N.S. Romanovski , On torsion in factors of polynilpotent series of a group with a single relation, International Jour nal of Algebra and Computation, 14, No. 4 (2004), 513-523. 18

work page 2004

[1] [1]

Hodges , Model Theory, Cambridge, Cambridge Univ

W. Hodges , Model Theory, Cambridge, Cambridge Univ. Press, 1985

work page 1985

[2] [2]

Scott, Algebraically closed groups, Proc

W.R. Scott, Algebraically closed groups, Proc. Amer. Math. Soc., 2 (1951), 118–121

work page 1951

[3] [3]

Myasnikov, V

A. Myasnikov, V. Roman ’kov , Verbally closed subgroups of free groups, J. Group Theory, 17, No. 1 (2014), 29–40

work page 2014

[4] [4]

G. M. Bergman , Supports of derivations, free factorizations and ranks of ﬁxed subgroups in free groups, Trans. Amer. Math. Soc., 351 (1999), 1551– 1573

work page 1999

[5] [5]

Unsolved problems in group theor y

The Kourovka Notebook. Unsolved problems in group theor y. No. 19 (2018),Editors: E.I. Khukhro and V.D. Mazurov. Sobolev Ins titute of Math. Siberian Branch RAS, Novosibirsk

work page 2018

[6] [6]

Baumslag, A

G. Baumslag, A. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory, Contemporary Math., 296 (2002), 1–38

work page 2002

[7] [7]

Retracts of free groups and a question of Bergman

I. Snopce, S. Tanushevski, P. Zalesskii , Retracts of free groups and a ques- tion of Bergman, arXiv:1902.02378 [math.GR]. Feb 6 (2019). 16

work page internal anchor Pith review Pith/arXiv arXiv 1902

[8] [8]

Roman ’kov, N.G

V.A. Roman ’kov, N.G. Khisamiev, Verbally and existentially closed sub- groups of free nilpotent groups. Algebra and Logic. 2013. 52, No. 4, 336–351

work page 2013

[9] [9]

A. M. Mazhuga , On free decompositions of verbally closed subgroups in fre e products of ﬁnite groups, J. Group Theory, 20, No. 5 (2017), 971–986

work page 2017

[10] [10]

Mazhuga , Strongly verbally closed groups, J

A.M. Mazhuga , Strongly verbally closed groups, J. Algebra, 493 (2018), 171–184

work page 2018

[11] [11]

Mazhuga , Verbally closed subgroups,Thesis, Lomonosov Moscow State University, Moscow, 2018 (In Russian)

A.M. Mazhuga , Verbally closed subgroups,Thesis, Lomonosov Moscow State University, Moscow, 2018 (In Russian)

work page 2018

[12] [12]

Ant. A. Klyachko, A.M. Mazhuga , Verbally closed virtually free subgroups, Sbornik: Mathematics, 209, No. 6 (2018), 850–856

work page 2018

[13] [13]

A.Klyachko, A

A. A.Klyachko, A. M. Mazhuga, V. Y. Miroshnichenko , Virtually free ﬁnite-normal-subgroup-free groups are strongly verbally closed, J. Algebra, 510 (2018), 319–330

work page 2018

[14] [14]

Timoshenko , Test elements and test rank of a free metabelian group, Siberian Mathematical Journal, 47, No

E.I. Timoshenko , Test elements and test rank of a free metabelian group, Siberian Mathematical Journal, 47, No. 6 (2000), 1200–1204

work page 2000

[15] [15]

Roman ’kov, Test elements for free solvable groups of rank 2

V.A. Roman ’kov, Test elements for free solvable groups of rank 2. Algebra and Logic. 2001. 40, No.2. 106–111

work page 2001

[16] [16]

Timoshenko , Computing test rank for a free solvable group

E.I. Timoshenko , Computing test rank for a free solvable group. Algebra and Logic, 45,No. 4 (2006), 254–260

work page 2006

[17] [17]

Gupta, E.I

C.K. Gupta, E.I. Timoshenko, Test rank for fome free polynilpotent Groups, Algebra and Logic 42, No. 1 (2003), 20–28

work page 2003

[18] [18]

Timoshenko , Endomorphisms and universal theories of solvable groups

E.I. Timoshenko , Endomorphisms and universal theories of solvable groups. Novosibirsk State Technical University, Novosibi rsk, 2011 (In Rus- sian)

work page 2011

[19] [19]

Remeslennikov, V.A

V.N. Remeslennikov, V.A. Sokolov , Some properties of Magnus embedding, Algebra and Logic, 9 (1970), 342–349

work page 1970

[20] [20]

Crowell, R.H

R.H. Crowell, R.H. Fox , Introduction to knot theory, Springer-Verlag, Berlin-Heidelberg- New York, 1963

work page 1963

[21] [21]

V. A. Roman ’kov, Essays in algebra and cryptology. Solvable groups, Omsk, Omsk State University Publishing House, 2017. 17

work page 2017

[22] [22]

Baumslag , Some subgroup theorems for free υ-groups, Trans

G. Baumslag , Some subgroup theorems for free υ-groups, Trans. Amer. Math. Soc., 108 (1963), 516–525

work page 1963

[23] [23]

Roman ’kov, Equations in free metabelian groups, Siberian Mathe- matical Journal

V.A. Roman ’kov, Equations in free metabelian groups, Siberian Mathe- matical Journal. 1979. 20, No. 3 (1979). 469–471

work page 1979

[24] [24]

Neumann, Varieties of groups, Springer-Verlag, Berlin-Heidelber g, 1967

H. Neumann, Varieties of groups, Springer-Verlag, Berlin-Heidelber g, 1967

work page 1967

[25] [25]

Artamonov , Projective metabelian groups and Lie algebras, Uspekhi Mat

V.A. Artamonov , Projective metabelian groups and Lie algebras, Uspekhi Mat. Nauk, 32, No. 3(195) (1977), 166

work page 1977

[26] [26]

Lewin , A note on zero divisors in group rings, Proc

J. Lewin , A note on zero divisors in group rings, Proc. Amer. Math. Soc ., 31 (1972), 357–359

work page 1972

[27] [27]

C. K. Gupta, N.S. Romanovski , On torsion in factors of polynilpotent series of a group with a single relation, International Jour nal of Algebra and Computation, 14, No. 4 (2004), 513-523. 18

work page 2004