Sphicas48, A. Stakia, J. Steggemann, M. Stoye, M. Tosi, D. Treille, A. Tsirou,
V. Veckalns49, M. Verweij, W.D. Zeuner Paul Scherrer Institut, Villigen,
Switzerland
W. Bertl${}^{\textrm{\textdagger}}$, L. Caminada50, K. Deiters, W. Erdmann, R.
Horisberger, Q. Ingram, H.C. Kaestli, D. Kotlinski, U. Langenegger, T. Rohe,
S.A. Wiederkehr ETH Zurich - Institute for Particle Physics and Astrophysics
(IPA), Zurich, Switzerland
M. Backhaus, L. Bäni, P. Berger, B. Casal, N. Chernyavskaya, G. Dissertori, M.
Dittmar, M. Donegà, C. Dorfer, C. Grab, C. Heidegger, D. Hits, J. Hoss, T.
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Quittnat, M. Reichmann, D. Ruini, D.A. Sanz Becerra, M. Schönenberger, L.
Shchutska, V.R. Tavolaro, K. Theofilatos, M.L. Vesterbacka Olsson, R. Wallny,
D.H. Zhu Universität Zürich, Zurich, Switzerland
T.K. Aarrestad, C. Amsler51, D. Brzhechko, M.F. Canelli, A. De Cosa, R. Del
Burgo, S. Donato, C. Galloni, T. Hreus, B. Kilminster, I. Neutelings, D.
Pinna, G. Rauco, P. Robmann, D. Salerno, K. Schweiger, C. Seitz, Y. Takahashi,
A. Zucchetta National Central University, Chung-Li, Taiwan
V. Candelise, Y.H. Chang, K.y. Cheng, T.H. Doan, Sh. Jain, R. Khurana, C.M.
Kuo, W. Lin, A. Pozdnyakov, S.S. Yu National Taiwan University (NTU), Taipei,
Taiwan
P. Chang, Y. Chao, K.F. Chen, P.H. Chen, F. Fiori, W.-S. Hou, Y. Hsiung, Arun
Kumar, Y.F. Liu, R.-S. Lu, E. Paganis, A. Psallidas, A. Steen, J.f. Tsai
Chulalongkorn University, Faculty of Science, Department of Physics, Bangkok,
Thailand
B. Asavapibhop, K. Kovitanggoon, G. Singh, N. Srimanobhas Çukurova University,
Physics Department, Science and Art Faculty, Adana, Turkey
M.N. Bakirci52, A. Bat, F. Boran, S. Cerci53, S. Damarseckin, Z.S. Demiroglu,
C. Dozen, I. Dumanoglu, S. Girgis, G. Gokbulut, Y. Guler, I. Hos54, E.E.
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Ozdemir56, B. Tali53, U.G. Tok, S. Turkcapar, I.S. Zorbakir, C. Zorbilmez
Middle East Technical University, Physics Department, Ankara, Turkey
G. Karapinar57, K. Ocalan58, M. Yalvac, M. Zeyrek Bogazici University,
Istanbul, Turkey
I.O. Atakisi, E. Gülmez, M. Kaya59, O. Kaya60, S. Tekten, E.A. Yetkin61
Istanbul Technical University, Istanbul, Turkey
M.N. Agaras, S. Atay, A. Cakir, K. Cankocak, Y. Komurcu Institute for
Scintillation Materials of National Academy of Science of Ukraine, Kharkov,
Ukraine
B. Grynyov National Scientific Center, Kharkov Institute of Physics and
Technology, Kharkov, Ukraine
L. Levchuk University of Bristol, Bristol, United Kingdom
F. Ball, L. Beck, J.J. Brooke, D. Burns, E. Clement, D. Cussans, O. Davignon,
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Zenz Brunel University, Uxbridge, United Kingdom
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Institute of Technology, Pasadena, USA
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Tallahassee, USA
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Yohay Florida Institute of Technology, Melbourne, USA
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University of Kansas, Lawrence, USA
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State University of New York at Buffalo, Buffalo, USA
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B. Roozbahani Northeastern University, Boston, USA
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Northwestern University, Evanston, USA
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Columbus, USA
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Hart, C. Hill, W. Ji, T.Y. Ling, W. Luo, B.L. Winer, H.W. Wulsin Princeton
University, Princeton, USA
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A. Kalogeropoulos, D. Lange, J. Luo, D. Marlow, K. Mei, I. Ojalvo, J. Olsen,
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Puerto Rico, Mayaguez, USA
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R. Xiao, W. Xie Purdue University Northwest, Hammond, USA
T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA
Z. Chen, K.M. Ecklund, S. Freed, F.J.M. Geurts, M. Guilbaud, M. Kilpatrick, W.
Li, B. Michlin, B.P. Padley, J. Roberts, J. Rorie, W. Shi, Z. Tu, J. Zabel, A.
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Verzetti The Rockefeller University, New York, USA
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Jersey, Piscataway, USA
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R. Montalvo, K. Nash, M. Osherson, H. Saka, S. Salur, S. Schnetzer, D.
Sheffield, S. Somalwar, R. Stone, S. Thomas, P. Thomassen, M. Walker
University of Tennessee, Knoxville, USA
A.G. Delannoy, J. Heideman, G. Riley, K. Rose, S. Spanier, K. Thapa Texas A&M
University, College Station, USA
O. Bouhali74, A. Castaneda Hernandez74, A. Celik, M. Dalchenko, M. De Mattia,
A. Delgado, S. Dildick, R. Eusebi, J. Gilmore, T. Huang, T. Kamon75, R.
Mueller, Y. Pakhotin, R. Patel, A. Perloff, L. Perniè, D. Rathjens, A.
Safonov, A. Tatarinov Texas Tech University, Lubbock, USA
N. Akchurin, J. Damgov, F. De Guio, P.R. Dudero, J. Faulkner, E. Gurpinar, S.
Kunori, K. Lamichhane, S.W. Lee, T. Mengke, S. Muthumuni, T. Peltola, S.
Undleeb, I. Volobouev, Z. Wang Vanderbilt University, Nashville, USA
S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K.
Padeken, J.D. Ruiz Alvarez, P. Sheldon, S. Tuo, J. Velkovska, Q. Xu University
of Virginia, Charlottesville, USA
M.W. Arenton, P. Barria, B. Cox, R. Hirosky, M. Joyce, A. Ledovskoy, H. Li, C.
Neu, T. Sinthuprasith, Y. Wang, E. Wolfe, F. Xia Wayne State University,
Detroit, USA
R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa, S. Zaleski University
of Wisconsin - Madison, Madison, WI, USA
M. Brodski, J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, L. Dodd, S. Duric,
B. Gomber, M. Grothe, M. Herndon, A. Hervé, U. Hussain, P. Klabbers, A.
Lanaro, A. Levine, K. Long, R. Loveless, V. Rekovic, T. Ruggles, A. Savin, N.
Smith, W.H. Smith, N. Woods †: Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
3: Also at Universidade Estadual de Campinas, Campinas, Brazil
4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
5: Also at Universidade Federal de Pelotas, Pelotas, Brazil
6: Also at Université Libre de Bruxelles, Bruxelles, Belgium
7: Also at Institute for Theoretical and Experimental Physics, Moscow, Russia
8: Also at Joint Institute for Nuclear Research, Dubna, Russia
9: Also at Suez University, Suez, Egypt
10: Now at British University in Egypt, Cairo, Egypt
11: Also at Zewail City of Science and Technology, Zewail, Egypt
12: Also at Department of Physics, King Abdulaziz University, Jeddah, Saudi
Arabia
13: Also at Université de Haute Alsace, Mulhouse, France
14: Also at Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State
University, Moscow, Russia
15: Also at Tbilisi State University, Tbilisi, Georgia
16: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
17: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
18: Also at University of Hamburg, Hamburg, Germany
19: Also at Brandenburg University of Technology, Cottbus, Germany
20: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
21: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary
22: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary
23: Also at Indian Institute of Technology Bhubaneswar, Bhubaneswar, India
24: Also at Institute of Physics, Bhubaneswar, India
25: Also at Shoolini University, Solan, India
26: Also at University of Visva-Bharati, Santiniketan, India
27: Also at University of Ruhuna, Matara, Sri Lanka
28: Also at Isfahan University of Technology, Isfahan, Iran
29: Also at Yazd University, Yazd, Iran
30: Also at Plasma Physics Research Center, Science and Research Branch,
Islamic Azad University, Tehran, Iran
31: Also at Università degli Studi di Siena, Siena, Italy
32: Also at INFN Sezione di Milano-Bicocca a, Università di Milano-Bicocca b,
Milano, Italy
33: Also at International Islamic University of Malaysia, Kuala Lumpur,
Malaysia
34: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
35: Also at Consejo Nacional de Ciencia y Tecnología, Mexico city, Mexico
36: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
37: Also at Institute for Nuclear Research, Moscow, Russia
38: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
39: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
40: Also at University of Florida, Gainesville, USA
41: Also at P.N. Lebedev Physical Institute, Moscow, Russia
42: Also at California Institute of Technology, Pasadena, USA
43: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
44: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
45: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy
46: Also at University of Belgrade, Faculty of Physics and Vinca Institute of
Nuclear Sciences, Belgrade, Serbia
47: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
48: Also at National and Kapodistrian University of Athens, Athens, Greece
49: Also at Riga Technical University, Riga, Latvia
50: Also at Universität Zürich, Zurich, Switzerland
51: Also at Stefan Meyer Institute for Subatomic Physics (SMI), Vienna,
Austria
52: Also at Gaziosmanpasa University, Tokat, Turkey
53: Also at Adiyaman University, Adiyaman, Turkey
54: Also at Istanbul Aydin University, Istanbul, Turkey
55: Also at Mersin University, Mersin, Turkey
56: Also at Piri Reis University, Istanbul, Turkey
57: Also at Izmir Institute of Technology, Izmir, Turkey
58: Also at Necmettin Erbakan University, Konya, Turkey
59: Also at Marmara University, Istanbul, Turkey
60: Also at Kafkas University, Kars, Turkey
61: Also at Istanbul Bilgi University, Istanbul, Turkey
62: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom
63: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
64: Also at Monash University, Faculty of Science, Clayton, Australia
65: Also at Instituto de Astrofísica de Canarias, La Laguna, Spain
66: Also at Bethel University, St. Paul, USA
67: Also at Utah Valley University, Orem, USA
68: Also at Purdue University, West Lafayette, USA
69: Also at Beykent University, Istanbul, Turkey
70: Also at Bingol University, Bingol, Turkey
71: Also at Erzincan University, Erzincan, Turkey
72: Also at Sinop University, Sinop, Turkey
73: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
74: Also at Texas A&M University at Qatar, Doha, Qatar
75: Also at Kyungpook National University, Daegu, Korea


# On $p$-nonsingular systems of equations over solvable groups111 This work
was supported by the Russian Science Foundation, project no. 22-11-00075

Mikhail A. Mikheenko
Faculty of Mechanics and Mathematics of Lomonosov Moscow State University
Moscow Center for Fundamental and Applied Mathematics
<EMAIL_ADDRESS>
###### Abstract

Any group that has a subnormal series, in which all factors are abelian and
all except the last one are $p^{\prime}$-torsion-free, can be embedded into a
group with a subnormal series of the same length, with the same properties and
such that any $p$-nonsingular system of equations over this group is solvable
in this group itself. This helps us to prove that the minimal order of a
metabelian group, over which there is a unimodular equation that is unsolvable
in metabelian groups, is $42$.

##  0 Introduction

In this article $\mathbb{Z}_{p}$ for a prime $p$ denotes the field with $p$
elements, $(\mathbb{Z}_{p})^{*}$ denotes the group of its invertible elements.
A cyclic group of order $n$ is denoted as $\langle a\rangle_{n}$, $\langle
b\rangle_{n}$, $\langle g\rangle_{n}$ etc. An infinite cyclic group is denoted
as $\langle g\rangle_{\infty}$, $\langle x_{1}\rangle_{\infty}$ etc. The
Cartesian (or unrestriscted direct) product of groups $\\{G_{i}\\}_{i\in I}$
is denoted by $\prod\limits_{i\in I}G_{i}$, whereas their direct (restricted)
product is denoted as $\underset{i\in I}{\times}G_{i}$. The group ring of a
group $G$ with a coefficient ring $R$ is denoted by $RG$.

Here we call a group $G$ an extension of a group $A$ by a group $B$ if $G$ has
the normal subgroup $A$ and the quotient group $G/A$ is isomorphic to $B$. For
elements $g$ and $h$ of a group $G$ the expression $g^{h}$ denotes $h^{-1}gh$
and the expression $g^{nh}$, where $n$ is an integer (may be negative),
denotes $h^{-1}g^{n}h$. Also, $[g,h]$ denotes $g^{-1}h^{-1}gh$. The Cartesian
(or unrestricted) wreath product $A\bar{\wr}B$ is viewed as
$\left(\prod\limits_{b\in B}A_{b}\right)\leftthreetimes B$, in which $A_{b}$
are copies of $A$ and $B$ acts on $\prod\limits_{b\in B}A_{b}$ such that
$\left((a_{b})_{b\in B}\right)^{b_{1}}=(a_{b})_{bb_{1}\in
B}=(a_{bb_{1}^{-1}})_{b\in B}$ if $b_{1}\in B$ and $a_{b}\in A_{b}$ for any
$b\in B$.

The paper is devoted to equations and systems of equations over groups.

######  Definition.

An equation $w(x_{1},\ldots,x_{n})=1$ with coefficients from $G$ and variables
$x_{1},\ldots,x_{n}$ is called solvable over $G$ if $G$ can be embedded into
some group $\widetilde{G}\supset G$ which contains a solution of this equation
(i.e. there are elements $\widetilde{g}_{1},\ldots,\widetilde{g}_{n}$ such
that $w(\widetilde{g}_{1},\ldots,\widetilde{g}_{n})=1$). The group
$\widetilde{G}$ is called a solution group.

The solvability of a (finite or infinite) system of equations over a group is
defined likewise.

######  Definition.

Let $\\{w_{j}=1\\}_{j\in J}$ be a system of equations with variables
$\\{x_{i}\\}_{i\in I}$ over a group $G$. For each word $w_{j}$ construct a row
of exponent sums of all variables $x_{i}$ in the word $w_{j}$.

The system is called non-singular ($p$-nonsingular for a prime $p$,
unimodular) if these rows are linearly independent over $\mathbb{Q}$
($\mathbb{Z}_{p}$, over all the fields $\mathbb{Z}_{p}$ for every prime $p$
respectively).

In particular, an equation $w(x)=1$ in one variable is non-singular
($p$-nonsingular, unimodular) if the exponent sum of $x$ in $w$ is non-zero
(is not divisible by $p$, is equal to $\pm 1$ respectively).

Clearly, if a system of equations is $p$-nonsingular for some prime $p$, then
it is non-singular as well. Also, a unimodular system is the same as a
$p$-nonsingular system for every prime $p$, therefore, in particular,
unimodular systems are non-singular.

######  Example.

The system of equations

$\left\\{\begin{array}[]{l}[x,y]x^{2}g_{1}y^{-3}=1,\\\ [y,z]z=1,\\\
xg_{2}yg_{3}z=1\\\ \end{array}\right.$

has the following exponent sums rows organized into a matrix:

$\begin{pmatrix}2&-3&0\\\ 0&0&1\\\ 1&1&1\end{pmatrix}.$

The determinant of this matrix is $-5$, so the system is non-singular,
$2$-nonsingular, $3$-nonsingular, but is $5$-singular, so it is not
unimodular.

Non-singular systems of equations are solvable over groups from many classes.
For example, one of the classical results on equations over groups concerns
finite groups:

######  Theorem ([GR62]).

A finite non-singular system of equations over a finite group $G$ is solvable
over $G$. Moreover, a solution group can be chosen to be finite.

This theorem was repeatedly generalized, see [P08, KT17, T18, NT22, KM23].

The following conjecture is as yet neither proved nor disproved:

######  Howie conjecture ([How81]).

Any non-singular system of equations over any group is solvable over this
group.

In fact, if every finite non-singular system over a group $G$ is solvable over
$G$, then every infinite non-singular system over $G$ is solvable over $G$ as
well, for example, see [KMR23, Proposition 2]. It means that to check whether
a group satisfies the Howie conjecture, it suffices to consider only finite
systems. So, for example, the result from [GR62] states that the conjecture is
true for finite groups, though a solution group for an infinite system may not
be finite.

Besides finite groups, this conjecture is true for locally indicable groups as
well.

######  Definition.

A locally indicable group is a group whose any non-trivial finitely generated
subgroup admits a surjective homomorphism onto $\langle g\rangle_{\infty}$.

######  Definition.

A locally $p$-indicable group for a prime $p$ is a group whose any non-trivial
finitely generated subgroup admits a surjective homomorphism onto $\langle
g\rangle_{p}$.

Note that locally indicable groups are locally $p$-indicable for any prime
$p$. Also, it is easy to see that, if a group is locally $p$-indicable for an
arbitrarily large prime $p$, then it is locally indicable. This can be seen by
considering a quotient group of a finitely generated subgroup by the
commutator subgroup of this subgroup.

######  Theorem ([How81]).

A finite non-singular system of equations over a locally indicable group is
solvable over this group.

There is a similar result for $p$-nonsingular systems.

######  Theorem ([Kr85]).

Let $p$ be a prime. A finite $p$-nonsingular system of equations over a
locally $p$-indicable group is solvable over this group.

The following result shows that more is known about unimodular equations than
about non-singular ones.

######  Theorem ([K93]).

Any unimodular equation over a torsion-free group is solvable over this group.

This result was generalized in [K06]. It is unknown if an anologous result is
true for non-singular equations.

Some results are connected not only with the solvability of systems of
equations over groups but also with the solvability of systems in groups of
the same class. Aside from the already mentioned result from [GR62] for the
class of finite groups there is a long-ago obtained result for the class of
nilpotent groups.

######  Definition.

Let $p$ be a prime. A group $G$ is called $p^{\prime}$-torsion-free if any
non-identity element of $G$ has either infinite order or order equal to a
power of $p$.

######  Theorem ([Sh67]).

If

  * •

$G$ is a divisible nilpotent torsion-free group and there is a finite non-
singular system of equations over $G$;

  * •

$G$ is nilpotent and there is a finite unimodular system of equations over
$G$;

  * •

or $G$ is a $p^{\prime}$-torsion-free nilpotent group in which any element is
a $q$th power for any prime number $q$ not equal to $p$ (so $G$ is
$p^{\prime}$-divisible) and there is a finite $p$-nonsingular system of
equations over $G$;

then the system is solvable in $G$, and the solution in $G$ is unique.

This implies that, for example, any non-singular system over a nilpotent
torsion-free group is solvable in the divisible hull of the group, which is a
nilpotent group of the same nilpotency class.

[KMR23] studied a similar question for the class of solvable groups: when a
non-singular system over a solvable group has a solution in a solvable group
as well and what can be said about derived length of the solution group?

######  Proposition ([KMR23]).

There exists a metabelian group $G$ with a unimodular equation $w(x)=1$ over
it such that the equation $w(x)=1$ has no solutions in metabelian groups.

Moreover, $G$ can be chosen to be:

  * •

either finite (of order $42$)

  * •

or torsion-free and such that any unimodular equation (and even any non-
singular system) over $G$ has a solution in the class of solvable groups of
derived length $3$.

######  Theorem ([KMR23]).

Let a group $G$ have a subnormal series

$G=G_{1}\triangleright G_{2}\triangleright\ldots\triangleright
G_{n}\triangleright G_{n+1}=\\{1\\}$

in which all factors are abelian and all factors except the last one are
torsion-free. Then any (finite or infinite) non-singular system of equations
over $G$ has a solution in some group $\widetilde{G}\supset G$ which also has
a subnormal series

$\widetilde{G}=\widetilde{G}_{1}\triangleright\widetilde{G}_{2}\triangleright\ldots\triangleright\widetilde{G}_{n}\triangleright\widetilde{G}_{n+1}=\\{1\\}$

with abelian factors, and all factors of the series of $\widetilde{G}$, except
for the last factor, are torsion-free.

Section 2 of the present article contains the proof to a similar result for
$p$-nonsingular systems.

######  Main theorem.

Let $p$ be a prime number. Let a group $G$ have a subnormal series

$G=G_{1}\triangleright G_{2}\triangleright\ldots\triangleright
G_{n}\triangleright G_{n+1}=\\{1\\}$

with factors $B_{1},B_{2},\dots,B_{n}$, where $B_{i}=G_{i}/G_{i+1}$ are
abelian groups and all the factors $B_{i}$ except $B_{n}$ are
$p^{\prime}$-torsion-free.

Then there is a group $\widehat{G}\supset G$ with a subnormal series

$\widehat{G}=\widehat{G}_{1}\triangleright\widehat{G}_{2}\triangleright\ldots\triangleright\widehat{G}_{n}\triangleright\widehat{G}_{n+1}=\\{1\\}$

with the same properties (i.e. its length is $n$, factors are abelian and all
factors except the last one are $p^{\prime}$-torsion-free) such that any
$p$-nonsingular system of equations over $G$ has a solution in $\widehat{G}$.
Moreover, every $p$-nonsingular system over $\widehat{G}$ is solvable in
$\widehat{G}$ itself.

Arguments of Section 2 allow us to strengthen the theorem from [KMR23].

Lemma 1 from Section 1 plays an important role in the proof of the main
theorem. Also, the method of the proof is based on methods that were developed
in [NT22, KM23, KMR23].

In Section 3 the main theorem helps us to prove the following fact.

######  Proposition 1.

The metabelian group $\begin{pmatrix}1&\mathbb{Z}_{7}\\\
0&(\mathbb{Z}_{7})^{*}\\\ \end{pmatrix}$ of order $42$ from [KMR23] is a
minimal (by order) example of a metabelian group over which there exists a
unimodular equation that has no solutions in metabelian groups.

Some open questions on systems of equations over solvable groups are
formulated in Section 4.

The author thanks the Theoretical Physics and Mathematics Advancement
Foundation ‘‘BASIS’’. The author also thanks Anton A. Klyachko for valuable
advices and remarks.

##  1 Images of rows of group ring elements

######  Proposition 2.

Let $R$ be an associative ring with unity of characteristic $p$. Let $M\in
R[x]/(x^{p^{k}}-1)$ be an element such that $f(M)$ is not a left (right) zero
divisor, where $f\colon R[x]/(x^{p^{k}}-1)\to R$ is the natural ring
homomorphism with $f(x)=1.$ Then $M$ itself is not a left (right) zero
divisor.

###### Proof.

We consider the ‘‘left’’ case.

Consider $M$ as a polynomial in $(x-1)$ with coefficients from $R$:

$M=M_{0}+M_{1}(x-1)+M_{2}(x-1)^{2}+\ldots+M_{p^{k}-1}(x-1)^{p^{k}-1}.$

$(x-1)^{p^{k}}=x^{p^{k}}-1=0$ because the characteristic of $R$ is $p$. By the
definition of $f$, $M_{0}$ is equal to $f(M)$, hence, $M_{0}$ is not a left
zero divisor.

Now assume that $M$ is a left zero divisor. This means that there is a non-
zero polynomial $B=B_{0}+B_{1}(x-1)+\ldots+B_{p^{k}-1}(x-1)^{p^{k}-1}$ such
that $MB=0$. Consider $MB$ as a polynomial in $(x-1)$ as well. Look at the
constant term of $MB$. On the one hand, it is equal to $M_{0}B_{0}$, on the
other hand, it must be zero. $M_{0}$ is not a left zero divisor, therefore,
$B_{0}=0$. Then look at the linear term of $MB$. As $B_{0}$ equals $0$, the
linear term is equal to $M_{0}B_{1}(x-1)$, while it must be equal to $0$ as
well as the constant term. And so on. Eventually we get that all the
coefficients $B_{i}$ are equal to $0$. This contradicts the condition $B\neq
0$, thus, the assumption that $M$ is a left zero divisor is false.

The ‘‘right’’ case can be treated similarly. ∎

######  Corollary 1.

Let $R$ be an associative ring with unity of characteristic $p$. Let $M\in
R[x_{1},\ldots,x_{l}]/(x_{1}^{p^{k_{1}}}-1,\ldots,x_{l}^{p^{k_{l}}}-1)$ be an
element such that $f(M)$ is not a left (right) zero divisor, where $f\colon
R[x_{1},\ldots,x_{l}]/(x_{1}^{p^{k_{1}}}-1,\ldots,x_{l}^{p^{k_{l}}}-1)\to R$
is the natural ring homomorphism with $f(x_{i})=1.$ Then $M$ itself is not a
left (right) zero divisor.

###### Proof.

Let us use induction by $l$.

If $l=1$, then this is exactly Proposition 2.

Now show that, if the corollary is true for $l$ variables, then it is true for
$l+1$ variables as well.

Note that

$R[x_{1},\ldots,x_{l},x_{+1}]/(x_{1}^{p^{k_{1}}}-1,\ldots,x_{l}^{p^{k_{l}}}-1,x_{l+1}^{p^{k_{l+1}}}-1)=$
$=\left(R[x_{1},\ldots,x_{l}]/(x_{1}^{p^{k_{1}}}-1,\ldots,x_{l}^{p^{k_{l}}}-1)\right)[x_{l+1}]/(x_{l+1}^{p^{k_{l+1}}}-1).$

Denote
$R[x_{1},\ldots,x_{l}]/(x_{1}^{p^{k_{1}}}-1,\ldots,x_{l}^{p^{k_{l}}}-1)$ as
$Q$. The ring $Q$ has the characteristic $p$ as well. Look at the image $h(M)$
of $M$ under the natural homomorphism $h\colon
Q[x_{l+1}]/(x_{l+1}^{p^{k_{l+1}}}-1)\to Q$, $h(x_{l+1})=1$. This image is not
a left (right) zero divisor by induction. Then, by Proposition 2, the element
$M$ is also not a left (right) zero divisor. ∎

Now let $P$ be a finitely generated abelian $p$-group. It can be represented
as $P=\langle g_{1}\rangle_{p^{k_{1}}}\times\ldots\times\langle
g_{n}\rangle_{p^{k_{l}}}$. Then the group algebra $\mathbb{Z}_{p}P$ can be
represented as
$\mathbb{Z}_{p}P=\mathbb{Z}_{p}[x_{1},\ldots,x_{l}]/(x_{1}^{p^{k_{1}}}-1,\ldots,x_{l}^{p^{k_{l}}}-1)$.
And the ring of $n\times n$ matrices over $\mathbb{Z}_{p}P$ is
$M_{n}(\mathbb{Z}_{p})[x_{1},\ldots,x_{l}]/(x_{1}^{p^{k_{1}}}-1,\ldots,x_{l}^{p^{k_{l}}}-1)$.
Knowing that a non-singular matrix over a field is not a zero divisor, we get
the following fact.

######  Corollary 2.

Let $M\in M_{n}(\mathbb{Z}_{p}P)$ be an $n\times n$ matrix over the group
algebra, where $P$ is a finitely generated abelian $p$-group. Suppose that the
image of $M$ under the natural ring homomorphism $\varepsilon\colon
M_{n}(\mathbb{Z}_{p}P)\to M_{n}(\mathbb{Z}_{p})$, which sends the elements of
the group $P$ to the unity of $\mathbb{Z}_{p}$, is a non-singular matrix.

Then $M$ is not a zero divisor. In particular, the rows of $M$ are independent
over $\mathbb{Z}_{p}P$, i.e. there is no combination of these rows with
coefficients from $\mathbb{Z}_{p}P$ which is equal to zero, except the
combination where all the coefficients are zero.

The statement of Corollary 2 is also true for $P\times A$, where $P$ and $A$
are finitely generated abelian groups, $P$ is a $p$-group and $A$ is torsion-
free.

######  Corollary 3.

Let $M\in M_{n}(\mathbb{Z}_{p}\left(P\times A\right))$ be an $n\times n$
matrix over the group algebra, where $P$ is a finitely generated abelian
$p$-group and $A$ is a finitely generated torsion-free abelian group. Suppose
that the image of $M$ under the natural ring homomorphism $\varepsilon\colon
M_{n}(\mathbb{Z}_{p}\left(P\times A\right))\to M_{n}(\mathbb{Z}_{p})$, which
sends the elements of $P\times A$ to the unity of $\mathbb{Z}_{p}$, is a non-
singular matrix.

Then $M$ is not a zero divisor. In particular, the rows of $M$ are independent
over $\mathbb{Z}_{p}\left(P\times A\right)$.

###### Proof.

Look at the image of $M$ under the ring homomorphism $f\colon
M_{n}(\mathbb{Z}_{p}\left(P\times A\right))\to M_{n}(\mathbb{Z}_{p}A)$, which
sends the elements of $P$ to the unity of $\mathbb{Z}_{p}$ and does not change
the elements of $A$. The matrix $f(M)$ is not a zero divisor.

Indeed, $\mathbb{Z}_{p}A$ can be embedded into its field of fractions, as
$\mathbb{Z}_{p}A$ is an integral domain. Therefore, the matrix $f(M)$ not
being a zero divisor is equivalent to the non-singularity of $f(M)$ over the
field of fractions, i.e. to it having non-zero determinant. And the
determinant of $f(M)$ is not equal to zero, as by the condition the
determinant of $\varepsilon(M)=h\left(f(M)\right)$ is not equal to zero, where
$h\colon M_{n}(\mathbb{Z}_{p}A)\to M_{n}(\mathbb{Z}_{p})$ is the natural ring
homomorphism which sends the elements of $A$ to the unity of $\mathbb{Z}_{p}$.

Futher note that $M_{n}(\mathbb{Z}_{p}\left(P\times A\right))$ is naturally
isomorphic to the ring
$M_{n}(\mathbb{Z}_{p}A)[x_{1},\ldots,x_{l}]/(x_{1}^{p^{k_{1}}}-1,\ldots,x_{l}^{p^{k_{l}}}-1)$.
Then, by Corollary 1, in which $R$ is $M_{n}(\mathbb{Z}_{p}A)$, we get that
$M$ is not a zero divisor as $f(M)$ is not a zero divisor. ∎

The structure $P\times A$ (where $P$ and $A$ are from Corollary 3) is the
structure of every finitely generated $p^{\prime}$-torsion-free abelian group.
We can now go from the finitely generated case to the infintely generated one.

######  Corollary 4.

Let $M\in M_{n}(\mathbb{Z}_{p}D)$ be an $n\times n$ matrix over the group
algebra, where $D$ is a $p^{\prime}$-torsion-free abelian group. Suppose that
the image of $M$ under the natural ring homomorphism $\varepsilon\colon
M_{n}(\mathbb{Z}_{p}D)\to M_{n}(\mathbb{Z}_{p})$ is a non-singular matrix.

Then $M$ is not a zero divisor. In particular, the rows of $M$ are independent
over $\mathbb{Z}_{p}D$.

###### Proof.

Assume that this is not true. It means that there is a non-zero matrix $B$
over $\mathbb{Z}_{p}D$ such that either $BM=0$ or $MB=0$.

Look at all the elements of $B$ and $M$. There is a finite number of these
elements. Each of them contains a finite number of the elements of $D$ with a
non-zero coefficient. Hence, there is only a finite number of the elements of
$D$ which are contained in these matrices with a non-zero coefficient.

Consider the subgroup $H$ generated by those elements. It is a finitely
generated $p^{\prime}$-torsion-free abelian group. The elements of $B$ and $M$
are contained in the corresponding subalgebra $\mathbb{Z}_{p}H$. Therefore, we
get that $M$ is a zero divisor in a matrix ring over the group algebra of a
finitely generated $p^{\prime}$-torsion-free abelian group, which is
prohibited by Corollary 3.

It means that the assumption of $M$ being a zero divisor is false. ∎

Corollary 4 can be generalized to the infinite rows case.

######  Lemma 1.

Let $D$ be a $p^{\prime}$-torsion-free abelian group and $\\{m_{i}\\}_{i\in
I}$ be elements of a free module over $\mathbb{Z}_{p}D$ (i.e. rows of elements
of this group algebra). Consider the natural ring homomorphism
$f\colon\mathbb{Z}_{p}D\to\mathbb{Z}_{p}$ which sends the elements of the
group $D$ to the unity of $\mathbb{Z}_{p}$. This mapping naturally defines a
mapping from the free module over $\mathbb{Z}_{p}D$ to the vector space over
$\mathbb{Z}_{p}$.

Now let the system of rows $\\{f(m_{i})\\}_{i\in I}$ be linearly independent
over $\mathbb{Z}_{p}$. Then the system $\\{m_{i}\\}_{i\in I}$ is independent
over $\mathbb{Z}_{p}D$.

###### Proof.

Assume that $\\{m_{i}\\}_{i\in I}$ is dependent over $\mathbb{Z}_{p}D$. It
means that some finite subsystem of these rows is dependent. Without loss of
generality, suppose that this subsystem consists of
$m_{1},m_{2},\ldots,m_{n}$.

By the condition of the lemma $f(m_{1}),f(m_{2}),\ldots f(m_{n})$ are linearly
independent over $\mathbb{Z}_{p}$. So if we write the rows
$f(m_{1}),f(m_{2}),\ldots f(m_{n})$ one above another and remove all the zero
columns (after this only a finite number of columns remains), we get a matrix
with linearly independent rows. In particular, this matrix has a non-singular
square submatrix of size $n\times n$.

Now consider the corresponding submatrix for $m_{1},m_{2},\ldots m_{n}$. By
Corollary 4 the rows of this submatrix are independent over $\mathbb{Z}_{p}D$.
So the rows $m_{1},m_{2},\ldots m_{n}$ are independent over $\mathbb{Z}_{p}D$.
We get a contradiction, thus, the assumption of the dependence of
$\\{m_{i}\\}_{i\in I}$ is false. ∎

##  2 The proof of the main theorem

Let us use induction by $n$.

If $n=1$ (i.e. the group $G$ is abelian), then the divisible hull of $G$ can
be taken as $\widehat{G}$. Indeed, suppose that $\\{w_{j}=1\\}_{j\in J}$ is a
non-singular (even not necessarily $p$-nonsingular) system of equations over a
divisible abelian group $H$ in variables $\\{x_{i}\\}_{i\in I}$. $H$ can be
naturally embedded into the group $H\times\left(\underset{i\in
I}{\times}\langle x_{i}\rangle_{\infty}\right)/\langle\\{w_{j}\\}_{j\in
J}\rangle$ (the natural mapping of $H$ is injective because of the non-
singularity of $\\{w_{j}=1\\}_{j\in J}$), which, obviously, has a solution of
the system. Since $H$ is divisible, it is a direct factor. The image of
$\\{x_{i}\\}_{i\in I}$ under the projection onto $H$ forms a solution of the
system in $H$.

Now assume that $n>1$. Embed $G$ into the Cartesian wreath product
$G_{2}\bar{\wr}B_{1}$ by the Kaloujnine-Krasner theorem [KK51]. After this we
can embed $B_{1}$ into its divisible hull $\widehat{B}_{1}$ and embed $G_{2}$
(by induction) into a group $\widetilde{G}_{2}$ with a required subnormal
series of length $n-1$ and such that every $p$-nonsingular system of equations
over $\widetilde{G}_{2}$ has a solution in $\widetilde{G}_{2}$ itself. Thus,
we embed $G_{2}\bar{\wr}B_{1}$ into the wreath product
$\widetilde{G}_{2}\bar{\wr}\widehat{B}_{1}$. Note that the divisible hull
$\widehat{B}_{1}$ of the $p^{\prime}$-torsion-free abelian group $B_{1}$ is
$p^{\prime}$-torsion-free as well (to understand this, it suffices to consider
the torsion-free component and the $p$-primary component of $\widehat{B}_{1}$
and to see that the projection of $\widehat{B}_{1}$ onto the product of these
two components is injective on $B_{1}\subset\widehat{B}_{1}$).

Now we need the following lemma.

######  Lemma 2.

Suppose that $\widehat{H}$ is a group such that every $p$-nonsingular system
over it has a solution in $\widehat{H}$ itself. Also suppose that
$\widehat{B}$ is a divisible $p^{\prime}$-torsion-free abelian group.

Then every $p$-nonsingular system of equations over the Cartesian wreath
product
$\widehat{H}\bar{\wr}\widehat{B}=\left(\prod\limits_{b\in\widehat{B}}\widehat{H}_{b}\right)\leftthreetimes\widehat{B}$
has a solution in $\widehat{H}\bar{\wr}\widehat{B}$ itself.

###### Proof.

Suppose that $\\{v_{j}=1\\}_{j\in J}$ is a $p$-nonsingular system of equations
over $\widehat{H}\bar{\wr}\widehat{B}$ in variables $\\{x_{i}\\}_{i\in I}$.
Consider the image of this system under the natural mapping of coefficients
$\widehat{H}\bar{\wr}\widehat{B}\to\widehat{B}$. This image is a
$p$-nonsingular system of equations over the divisible abelian group
$\widehat{B}$, therefore, it has a solution in $\widehat{B}$ itself.

It means that we can change the variables in the initial system
$\\{v_{j}=1\\}_{j\in J}$ over $\widehat{H}\bar{\wr}\widehat{B}$ such that all
the words $v_{j}$ become words in the alphabet $\\{x_{i}^{\pm b}\\}_{i\in
I,b\in\widehat{B}}\bigsqcup\prod\limits_{b\in\widehat{B}}\widehat{H}_{b}$.
From now we assume that all the words $v_{j}$ were originally in this
alphabet. Denote $\prod\limits_{b\in\widehat{B}}\widehat{H}_{b}$ by $C$.

Let us introduce new variables $\\{x_{ib}\\}_{i\in I,b\in\widehat{B}}$ and
rewrite the words $v_{j}$ of coefficients from $C$ and the variables
$\\{x^{\pm 1}_{i}\\}$ as words $w_{j}$ of coefficients from $C$ and the
variables $\\{x^{\pm 1}_{ib}\\}$ such that for every $j\in J$ $v_{j}$ is
$w_{j}$ in which $x^{\pm 1}_{ib}$ is replaced by $x_{i}^{\pm b}$ for all $i\in
I$ and $b\in\widehat{B}$.

From now we look at $x_{i}$ as elements from $C$ and try to find the solution
of $\\{v_{j}=1\\}_{j\in J}$ among these elements. Denote the coordinate of
$x_{i}$, correspoding to the factor $\widehat{H}_{b}$, as $[x_{i}]_{b}$. By
the action of $\widehat{B}$ on the base of the wreath product we get that for
every $d\in\widehat{B}$

$[x_{i}^{d}]_{b}=[x_{i}]_{bd^{-1}}.$

Now find the coordinates of the word
$v_{j}=w_{j}(c_{1},\ldots,c_{k},x_{i_{1}}^{b_{1}},\ldots,x_{i_{l}}^{b_{l}})$,
where $c_{1},\ldots,c_{k}$ are the coefficients of the equation:

$[v_{j}]_{b}=w_{j}([c_{1}]_{b},\ldots,[c_{k}]_{b},[x_{i_{1}}^{b_{1}}]_{b},\ldots,[x_{i_{l}}^{b_{l}}]_{b})=$
$=w_{j}([c_{1}]_{b},\ldots,[c_{k}]_{b},[x_{i_{1}}]_{bb_{1}^{-1}},\ldots,[x_{i_{l}}]_{bb_{l}^{-1}}).$

Thereby finding a solution $\\{x_{i}\\}_{i\in I}$ of the system
$\\{v_{j}=1\\}_{j\in J}$ in the group
$C=\prod\limits_{b\in\widehat{B}}\widehat{H}_{b}$ is the same as finding a set
of elements $\\{[x_{i}]_{b}\\}_{i\in I,b\in B}$ of the group $\widehat{H}$
such that

$w_{j}([c_{1}]_{b},\ldots,[c_{k}]_{b},[x_{i_{1}}]_{bb_{1}^{-1}},\ldots,[x_{i_{l}}]_{bb_{l}^{-1}})=1$

for any $j\in J$ and $b\in\widehat{B}$.

Now for every $i\in I$ and $b\in\widehat{B}$ we consider $[x_{i}]_{b}$ as the
variable $y_{i,b}$. It suffices to show that the system $\\{f_{j,b}=1\\}$,
where
$f_{j,b}=w_{j}([c_{1}]_{b},\ldots,[c_{k}]_{b},y_{i_{1},bb_{1}^{-1}},\ldots,y_{i_{l},bb_{l}^{-1}})$,
is $p$-nonsingular in the variables $\\{y_{i,b}\\}_{i\in I,b\in\widehat{B}}$.
Then, by the condition of the lemma, this system is solvable in $\widehat{H}$,
and that finishes the proof of the lemma. To show this, ‘‘forget’’ about the
coefficients from $C$ and look only at exponent sums of the variables
$y_{i,b}$ in the words $f_{j,b_{1}}$.

For now fix an index $j\in J$. For every $b\in\widehat{B}$ suppose that the
exponent sum of $y_{i,b}$ in $f_{j,b_{1}}$ is $n_{i,b}$ modulo $p$. Then for
each word $f_{j,b_{1}}$ and index $i\in I$, consider the element
$\sum\limits_{b\in\widehat{B}}n_{i,b}b$ of $\mathbb{Z}_{p}\widehat{B}$. It is
easy to see that this element is equal to the similar element for $f_{j,1}$
multiplied by $b_{1}$. After repeating this process for every index $i\in I$,
for the word $f_{j,b_{1}}$ we construct a row $m_{j,b_{1}}$ of elements of the
group algebra, in which the $i$th element corresponds to the index $i$. And
again, it is $m_{j,1}$ multiplied by $b_{1}$. A linear combination of these
rows for words $f_{j,b}$ is just $m_{j,1}$ multiplied by an element of
$\mathbb{Z}_{p}\widehat{B}$ (as it is a linear combination of rows
$m_{j,b}=b\cdot m_{j,1}$). Also, a linear combination of the rows (with the
elements from the group algebra) $m_{j,b}$ is equal to zero if and only if the
same combination of the rows of the variables exponent sums modulo $p$ (so the
elements of these rows are elements of $\mathbb{Z}_{p}$) in the words
$f_{j,b}$ is equal to zero as well.

Now the index $j$ is not fixed. Look at some linear combination $\lambda$ of
the rows $m_{j,b}$. It is a sum of linear combinations of the rows having the
same index $j$. Each of these combinations looks like
$(n_{1}b_{1}+\ldots+n_{t}b_{t})m_{j,1}$. So $\lambda$ is a combination of the
rows $m_{j,1}$ with coefficients from $\mathbb{Z}_{p}\widehat{B}$ for some
indices $j\in J$. Again, this combination is equal to zero if and only if the
same combination of the exponent sums rows modulo $p$ for the words $f_{j,b}$
is equal to zero. In particular, the $p$-nonsingularity of $\\{f_{j,b}=1\\}$
is equivalent to the independence of the rows $\\{m_{j,1}\\}_{j\in J}$ over
$\mathbb{Z}_{p}\widehat{B}$.

We can consider these rows as elements of a free module over
$\mathbb{Z}_{p}\widehat{B}$. Then look at the images of the rows $m_{j,1}$
under the natural ring homomorphism
$\varepsilon\colon\mathbb{Z}_{p}\widehat{B}\to\mathbb{Z}_{p}$ which sends the
elements of $\widehat{B}$ to the unity of $\mathbb{Z}_{p}$. The image of the
element from the $i$th place in the row $m_{j,1}$ is equal to the sum modulo
$p$ of the exponent sums of $y_{i,b}$ in $f_{j,1}$ by all $b\in\widehat{B}$
while the index $i$ is fixed. In other words, it is the sum for all
$b\in\widehat{B}$ of exponent sums of $x_{i,b}$ (index $i$ is fixed) in
$w_{j}$ modulo $p$. And this is the same as the exponent sum of $x_{i}$ in the
word $v_{j}$ modulo $p$. Therefore, the images of the rows $m_{j,1}$ for all
$j\in J$ are linearly independent over $\mathbb{Z}_{p}$. Hence, by Lemma 1,
the rows $m_{j,1}$ for all $j\in J$ are independent over
$\mathbb{Z}_{p}\widehat{B}$.

Thus, $\\{f_{j,b}=1\\}$ is $p$-nonsingular. It means that this system is
solvable in $\widehat{H}$, which finishes the proof of the lemma. ∎

After Lemma 2 is proved, it can be applied to
$\widetilde{G}_{2}\bar{\wr}\widehat{B}_{1}$ to get that any $p$-nonsingular
system of equations over this group is solvable in
$\widetilde{G}_{2}\bar{\wr}\widehat{B}_{1}$ itself. As this group contains
$G$, we can take $\widetilde{G}_{2}\bar{\wr}\widehat{B}_{1}$ as $\widehat{G}$.
The theorem’s conclusion is true for $\widehat{G}$: it has a required
subnormal series of length $n$ and any $p$-nonsingular system of equations
over $\widehat{G}$ is solvable in $\widehat{G}$ itself.

The main theorem is now proved. ∎

######  Remark 1.

If in the condition of the main theorem $n$ (the length of subnormal series)
is also equal to the derived length of $G$, then every $p$-nonsingular system
of equations over $G$ has a solution in a solvable group of the same derived
length as $G$.

######  Remark 2.

An analogue of Lemma 1 in the case where $D$ is a torsion-free abelian group
and $\mathbb{Z}_{p}$ is replaced by $\mathbb{Q}$ is true as well.

Therefore, if we repeat the proof of the main theorem, we can strengthen the
result from [KMR23]: for any group $G$ with a subnormal series having torsion-
free abelian factors (the last factor can have torsion) there is a group
$\widetilde{G}\supset G$ with a similar subnormal series of the same length
such that $\widetilde{G}$ contains a solution of every non-singular system of
equations over $G$ (and even over $\widetilde{G}$).

In [KMR23] each non-singular system over $G$ could have a different solution
group.

######  Remark 3.

An analogous result for two primes $p$ and $q$ (i.e. the main theorem for
$\\{p,q\\}^{\prime}$-torsion-free groups — these are group whose elements’
finite orders are not divisible by primes not from $\\{p,q\\}$ — instead of
$p^{\prime}$-torsion-free ones and $\\{p,q\\}$-nonsingular systems — these are
sumiltaneously $p$-nonsingular and $q$-nonsingular systems — instead of
$p$-nonsingular ones) is not true. This is shown by the example of order $42$
from [KMR23, Proposition 1a]: the first factor of this group’s subnormal
series is $\\{2,3\\}^{\prime}$-torsion-free, the equation over the group is
unimodular, however, the equation has no solutions in metabelian groups.

Moreover, for every two prime numbers $p$ and $q$ we can construct an example
of a metabelian group with the $\\{p,q\\}^{\prime}$-torsion-free first factor
such that there is a unimodular equation over this group which has no
solutions in metabelian groups.

######  Example.

Let $p,q$ be two different prime numbers. Consider the metabelian group
$G=\langle c\rangle_{2}\wr(\langle a\rangle_{p}\times\langle b\rangle_{q})$.
Its order is $2^{pq}pq$. Note that $cc^{ab}=[c,ab]$ belongs to the commutator
subgroup of $G$.

As $p$ and $q$ are coprime, there are integers $n$ and $m$ such that
$np+mq=1$. Then consider the equation

$x^{n}\cdot x^{na}\cdot\ldots\cdot x^{na^{p-1}}\cdot x^{m}\cdot
x^{mb}\cdot\ldots\cdot x^{mb^{q-1}}=cc^{ab}$

over $G$. Since $np+mq=1$, this equation is unimodular. And its right-hand
side is an element of the commutator subgroup of $G$.

Assume that there is a metabelian group $\widetilde{G}\supset G$ with a
solution $\widetilde{x}\in\widetilde{G}$ of this equation. After taking the
quotient $\widetilde{G}/\widetilde{G}^{\prime}$ the equality

$\widetilde{x}^{n}\cdot\widetilde{x}^{na}\cdot\ldots\cdot\widetilde{x}^{na^{p-1}}\cdot\widetilde{x}^{m}\cdot\widetilde{x}^{mb}\cdot\ldots\cdot\widetilde{x}^{mb^{q-1}}=cc^{ab}$

transforms into $\widetilde{x}=1$. It means that $\widetilde{x}$ belongs to
the commutator subgroup of $\widetilde{G}$. As $\widetilde{G}$ is metabelian,
this implies that $\widetilde{x}$ commutes with its conjugates.

Using the notation $\widetilde{x}^{g}\widetilde{x}^{h}=\widetilde{x}^{g+h}$
for simplicity, we get

$(\widetilde{x}^{n}\cdot\widetilde{x}^{na}\cdot\ldots\cdot\widetilde{x}^{na^{p-1}}\cdot\widetilde{x}^{m}\cdot\widetilde{x}^{mb}\cdot\ldots\cdot\widetilde{x}^{mb^{q-1}})^{1+ab}=$
$=\widetilde{x}^{(n(1+b)(1+a+\ldots+a^{p-1})+m(1+a)(1+b+\ldots+b^{q-1}))}=$
$=(\widetilde{x}^{n}\cdot\widetilde{x}^{na}\cdot\ldots\cdot\widetilde{x}^{na^{p-1}}\cdot\widetilde{x}^{m}\cdot\widetilde{x}^{mb}\cdot\ldots\cdot\widetilde{x}^{mb^{q-1}})^{a+b}.$

However,

$(cc^{ab})^{1+ab}=c^{1+2ab+a^{2}b^{2}}\neq
c^{a+b+a^{2}b+ab^{2}}=(cc^{ab})^{a+b}.$

Thus, we get a contradiction: some combinations of conjugates to the left-hand
side of the equation are equal, though the same combinations of conjugates to
the right-hand side are not.

Therefore, the assumption of finding a solution in a metabelian group is
false.

######  Remark 4.

From the proof of the main theorem we can deduce that
$\widehat{G}_{n}/\widehat{G}_{n+1}$ is a Cartesian power of the divisible hull
of the group $B_{n}=G_{n}/G_{n+1}$. Therefore, if $B_{n}$ is
$p^{\prime}$-torsion-free (like the other factors of the series of $G$), then
$\widehat{G}_{n}/\widehat{G}_{n+1}$ is $p^{\prime}$-torsion-free as well.

More generally, $\widehat{G}_{n}/\widehat{G}_{n+1}$ has the same set of prime
orders of elements as $B_{n}$. For example, if $B_{n}$ is $\langle
b\rangle_{30}$, then $\widehat{G}_{n}/\widehat{G}_{n+1}$ has elements of prime
orders $2$, $3$ and $5$ and has no elements of prime orders $7$, $13$ or $19$.

##  3 The proof of minimality

In this section we call a solvable group, over which there is a unimodular
equation that has no solutions in solvable groups of the same derived length,
a counterexample. ‘‘Counterexample’’ here means a counterexample to the
statement ‘‘any unimodular equation over a solvable group has a solution in
solvable groups with the same derived length’’.

The main theorem shows that, if a metabelian group $G$ is an extension of an
abelian group by an abelian $p$-group for some prime $p$, then any unimodular
equation over $G$ has a solution in some metabelian group, as unimodular
equations are $p$-nonsingular for every prime $p$. If, moreover, $G$ is
abelian, then any unimodular equation has a solution in $G$ itself. So in this
case $G$ can not be counterexamples.

This helps us to prove that every metabelian group of order at most $41$ is
not a counterexample and thus the group from Proposition 1 is indeed a minimal
counterexample by order.

Firstly, we prove two lemmas which reduce the amount of group orders to be
considered.

######  Lemma 3.

Suppose that a group $G$ has order $p^{k}$, where $p$ is a prime number. Then
any unimodular equation over $G$ has a solution in $G$ itself.

In particular, a group of order $p^{k}$ can not be a counterexample.

###### Proof.

$G$ is nilpotent, so Shmel’kin’s theorem [Sh67] can be applied. ∎

######  Lemma 4.

Suppose that a group $G$ has order $pq$, where $p$ and $q$ are different prime
numbers. Then:

  1. 1.

$G$ is metabelian.

  2. 2.

$G$ is an extension of an abelian group by an abelian $p$\- or $q$-group.

In particular, $G$ can not be a counterexample.

###### Proof.

Choose the greater number between $p$ and $q$. Without loss of generality,
suppose that it is $q$.

$G$ has a unique Sylow $q$-subgroup $\langle g\rangle_{q}$. As $\langle
g\rangle_{q}$ is the unique Sylow $q$-subgroup, it is normal in $G$,
therefore, $G$ is an extension of $\langle g\rangle_{q}$ by $\langle
h\rangle_{p}$. ∎

By Lemma 3 it follows that groups of orders $1$, $2$, $3$, $4$, $5$, $7$, $8$,
$9$, $11$, $13$, $16$, $17$, $19$, $23$, $25$, $27$, $29$, $31$, $32$, $37$
and $41$ can not be counterexamples.

Also, Lemma 4 shows that groups of orders $6$, $10$, $14$, $15$, $21$, $22$,
$26$, $33$, $34$, $35$, $38$ and $39$ are not counterexamples as well.

Now consider the remaining metabelian groups (which have order $12$, $18$,
$20$, $24$, $28$, $30$, $36$ or $40$).

  * •

If the order is $12$: suppose that a group $G$ of order $12$ is metabelian.
Consider possible indices of $G^{\prime}$ in $G$.

    * –

If the index is $1$: this is impossible, as $G$ is solvable.

    * –

If the index is $2$: $G$ is an extension of the abelian group $G^{\prime}$ by
the abelian $2$-group $G/G^{\prime}$.

    * –

If the index is $4$: the same as for the index $2$.

    * –

If the index is $3$: $G$ is an extension of the abelian group $G^{\prime}$ by
the abelian $3$-group $G/G^{\prime}$.

    * –

If the index is $6$: consider a subgroup $\widetilde{A}$ of order $2$ in
$G/G^{\prime}$. $\widetilde{A}$ is normal, as $G/G^{\prime}$ is abelian. Then
the pre-image $A\subset G$ of $\widetilde{A}$ under taking the quotient $G\to
G/G^{\prime}$ is a normal subgroup of order $4$, i.e. $A$ is abelian and has
index $3$. So $G$ is an extension of the abelian group $A$ by the abelian
$3$-group $G/A$.

    * –

If the index is $12$: $G$ is abelian.

Thus, in all cases $G$ is an extension of an abelian group by an abelian
$p$-group and can not be a counterexample.

  * •

If the order is $18$: a group $G$ of order $18$ has a Sylow $3$-subgroup $H$.
$H$ is abelian, as its order is $9$, and normal in $G$, as its index is $2$.
So $G$ is an extension of the abelian group $H$ by the cyclic group of order
$2$ and can not be a counterexample.

  * •

If the order is $20$: a group $G$ of order $20$ has a unique Sylow
$5$-subgroup $\langle g\rangle_{5}$. Thus, $G$ is an extension of $\langle
g\rangle_{5}$ by a group of order $4$ (which is an abelian $2$-group) and can
not be a counterexample.

  * •

If the order is $24$: suppose that a group $G$ of order $24$ is metabelian.
Consider possible indices of $G^{\prime}$ in $G$.

    * –

If the index is $1$: this is impossible, as $G$ is solvable.

    * –

If the index is $2$: $G$ is an extension of the abelian group $G^{\prime}$ by
the abelian $2$-group $G/G^{\prime}$.

    * –

If the index is $4$: the same as for the index $2$.

    * –

If the index is $8$: the same as for the index $2$.

    * –

If the index is $3$: $G$ is an extension of the abelian group $G^{\prime}$ by
the abelian $3$-group $G/G^{\prime}$.

    * –

If the index is $6$: the quotient $G/G^{\prime}$ is isomorphic to $\langle
g\rangle_{6}$, whereas $G^{\prime}$ is isomorphic either to $\langle
h\rangle_{4}$ or to $V_{4}$.

Consider the action of $G$ on $G^{\prime}$ by conjugation. As $G^{\prime}$ is
abelian, $G^{\prime}$ acts trivially on itself. I.e. this action induces an
action of $G/G^{\prime}$ on $G^{\prime}$. In other words, there is a
homomorphism from $G/G^{\prime}\cong\langle g\rangle_{6}$ either to
$\operatorname{Aut}(\langle h\rangle_{4})\cong\langle b\rangle_{2}$ or to
$\operatorname{Aut}(V_{4})\cong S_{3}$. In both cases the homomorphism has a
non-trivial kernel (i.e. a subgroup of $G/G^{\prime}$ which acts trivially on
$G^{\prime}$). Since $G/G^{\prime}$ is cyclic, the kernel is also cyclic.
Denote a generator of the kernel by $\widetilde{c}$.

Consider some element $c\in G$ from the pre-image of $\widetilde{c}$ under
taking the quotient $G\to G/G^{\prime}$. The trivial action of $\widetilde{c}$
on $G^{\prime}$ means that $c$ commutes with the elements of $G^{\prime}$.
Hence, $C:=\langle c,G^{\prime}\rangle$ is an abelian subgroup of $G$. Also,
$C$ normal in $G$, as $C$ is the pre-image of
$\langle\widetilde{c}\rangle\triangleleft G/G^{\prime}$ under taking the
quotient. And the index of $C$ in $G$ is equal to the index of
$\langle\widetilde{c}\rangle$ in $G/G^{\prime}$, i.e. to $1$, to $2$ or to
$3$. Therefore, $G$ is an extension of the abelian group $C$ by an abelian
$p$-group.

    * –

If the index is $12$: $G^{\prime}$ has order $2$, so it is $\langle
g\rangle_{2}$. The action of $G$ on $G^{\prime}$ by conjugation is an action
by trivial automorphisms, as there are no other automorphisms of $\langle
g\rangle_{2}$. Therefore, $G^{\prime}$ is contained in the center of $G$.

Now consider $G/G^{\prime}$, which is an abelian group of order $12$. It has a
direct factor $\langle a\rangle_{3}$. Consider its pre-image $A\subset G$
under taking the quotient $G\to G/G^{\prime}$. The subgroup $A$ is normal in
$G$ by the correspondence theorem, and $G/A$ is an abelian $2$-group. Also,
$A$ is either abelian or a group of order $6$ and with the center of order $2$
(as $G^{\prime}$ is contained in both the center of $G$ and the subgroup $A$).
But in the second case $A$ has a cyclic quotient by its center, so $A$ is
abelian.

Thus, $G$ is an extension of the abelian group $A$ by an abelian $2$-group.

    * –

If the index is $24$: $G$ is abelian.

Therefore, in all cases $G$ is an extension of an abelian group by an abelian
$p$-group and can not be a counterexample.

  * •

If the order is $28$: a group $G$ of order $28$ has a unique Sylow
$7$-subgroup $\langle g\rangle_{7}$. So $G$ is an extension of $\langle
g\rangle_{7}$ by a group of order $4$ (which is an abelian $2$-group) and can
not be a counterexample.

  * •

If the order is $30$: a metabelian group $G$ of order $30$ has a Hall subgroup
$A$ of order $15$, as $G$ is solvable. $A$ is abelian because it has unique
Sylow $3$\- and $5$-subgroups. And $A$ has index $2$ in $G$, so $A$ is normal
in $G$. Therefore, we get that $G$ is an extension of the abelian group $A$ by
the cyclic group of order $2$ and can not be a counterexample.

  * •

If the order is $36$: suppose that a group $G$ of order $36$ is metabelian.
Consider possible indices of $G^{\prime}$ in $G$.

    * –

If the index is $1$: this is impossible, as $G$ is solvable.

    * –

If the index is $2$: $G$ is an extension of the abelian group $G^{\prime}$ by
the abelian $2$-group $G/G^{\prime}$.

    * –

If the index is $4$: the same as for the index $2$.

    * –

If the index is $3$: $G$ is an extension of the abelian group $G^{\prime}$ by
the abelian $3$-group $G/G^{\prime}$.

    * –

If the index is $6$: both $G^{\prime}$ and $G/G^{\prime}$ are cyclic of order
$6$. Consider the action of $G$ on $G^{\prime}$ by conjugation. $G^{\prime}$
acts trivially on itself, as $G^{\prime}$ is abelian. So the action of $G$
induces an action of $G/G^{\prime}$ on $G^{\prime}$, i.e. a homomorphism from
$G/G^{\prime}$ to $\operatorname{Aut}(G^{\prime})$. As
$\operatorname{Aut}(G^{\prime})=\operatorname{Aut}(\langle
g\rangle_{6})\cong\langle h\rangle_{2}$, an element $\widetilde{c}\in
G/G^{\prime}$ of order $3$ acts on $G^{\prime}$ trivially.

Consider some element $c\in G$ from the pre-image of the element
$\widetilde{c}$ under taking the quotient $G\to G/G^{\prime}$. The subgroup
$C:=\langle c,G^{\prime}\rangle$ of the group $G$ is abelian, as conjugation
by the element $c$ acts on $G^{\prime}$ trivially (i.e. the element $c$
commutes with the elements of $G^{\prime}$). Also, $C$ is the pre-image of
$\langle\widetilde{c}\rangle\triangleleft G/G^{\prime}$ under taking the
quotient $G\to G/G^{\prime}$, Therefore, $C$ is normal in $G$ by the
correspondence theorem and has index $2$.

Thus, $G$ is an extension of the abelian group $A$ by an abelian $2$-group.

    * –

If the index is $12$: $G/G^{\prime}$ has a direct factor $\langle
a\rangle_{3}$. Consider its pre-image $A\subset G$ under taking the quotient
$G\to G/G^{\prime}$. $A$ is a subgroup of order $9$, so $A$ is abelian. $A$ is
normal in $G$ by the correspondence theorem and $G/A$ is an abelian $2$-group
(of order $4$).

    * –

If the index is $9$: the same as for the index $3$.

    * –

If the index is $18$: $G/G^{\prime}$ has a direct factor $\langle
a\rangle_{2}$. Consider its pre-image $A\subset G$ under taking the quotient
$G\to G/G^{\prime}$. $A$ is a subgroup of order $4$, so $A$ is abelian. $A$ is
normal in $G$ by the correspondence theorem and $G/A$ is an abelian $3$-group
(of order $9$).

    * –

If the index is $36$: $G$ is abelian.

Thus, in all cases $G$ is an extension of an abelian group by an abelian
$p$-group and can not be a counterexample.

  * •

If the order is $40$: a group $G$ of order $40$ has a unique Sylow
$5$-subgroup, so $G=\langle g\rangle_{5}\leftthreetimes B$, where $B$ is some
Sylow $2$-subgroup (of order $8$). Consider the action of $B$ on $\langle
g\rangle_{5}$ by conjugation in $G$. It is a homomorphism $\varphi$ from $B$
to $\operatorname{Aut}(\langle g\rangle_{5})\cong\langle h\rangle_{4}$. Since
the order of $B$ is greater than $4$, $\varphi$ has a non-trivial kernel, i.e.
a normal subgroup $\widetilde{C}\triangleleft B$ whose elements commute with
$g$ in $G$.

    * –

If $B$ is abelian, then $\widetilde{C}$ is also abelian.

    * –

If $B$ is $Q_{8}$, then $\widetilde{C}$ is either a proper subgroup of $Q_{8}$
(in this case $\widetilde{C}$ is abelian) or the whole group $Q_{8}$. In the
second case we can take any non-trivial proper subgroup of $Q_{8}$ as
$\widetilde{C}$ instead of the kernel of $\varphi$. Then $\widetilde{C}$ is an
abelian normal subgroup.

    * –

If $B$ is $D_{4}=\langle a\rangle_{4}\leftthreetimes\langle b\rangle_{2}$,
then consider the homomorphism $\varphi\colon D_{4}\to\langle h\rangle_{4}$.
As $\langle h\rangle_{4}$ is abelian it follows that
$\varphi(a)=\varphi(a^{b})=\varphi(a^{3})$. Then $\varphi(a^{2})=1$ and we can
take $\\{1,a^{2}\\}\triangleleft D_{4}$ as $\widetilde{C}$ instead of the
whole kernel of $\varphi$.

In all cases we get a subgroup $\widetilde{C}\triangleleft B$ which is a non-
trivial abelian normal subgroup of $B$ whose elements commute with $g$ in
$G=\langle g\rangle_{5}\leftthreetimes B$.

Then $C:=\langle\widetilde{C},g\rangle\subset G$ is an abelian subgroup of
$G$. Also, $C$ is normal in $G$, as $C$ is the pre-image of the normal
subgroup $\widetilde{C}$ under the natural homomorphism $G=\langle
g\rangle_{5}\leftthreetimes B\to B$. Finally, $B/\widetilde{C}\cong G/C$ is a
group of order $1$, $2$ or $4$, so $G/C$ is an abelian $2$-group. Thus, $G$ is
an extension of the abelian group $C$ by an abelian $2$-group and can not be a
counterexample.

As a result, a metabelian group of order at most $41$ can not be a
counterexample. With the existence of a counterexample of order $42$ we get
the proof of Proposition 1. ∎

##  4 Open questions

######  Question 1.

Is there any example of a solvable group of derived length greater than $2$
over which not every unimodular equation has a solution in solvable groups of
the same derived length?

######  Question 2.

In particular, is there an example of a solvable group of order less than $42$
but of derived length greater than $2$, over which not every unimodular
equation has a solution in solvable groups of the same derived length?

For example, is $S_{4}$ (which is a group of order $24$ and derived length
$3$) such an example? This group does not have a subnormal series satisfying
the condition of the main theorem.

######  Question 3.

Does every unimodular equation over the group from Proposition 1 have a
solution in a solvable group?

## References

  * [GR62] M. Gerstenhaber, O. S. Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. USA, 48:9 (1962), 1531-1533.
  * [How81] J. Howie, On pairs of 2-complexes and systems of equations over groups, J. Reine Angew Math., 1981:324 (1981), 165-174.
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multiple distinct notions when they do. For example: topology has one notion
of _dense subset of_ and, as discussed in Remark 11.5.1, when we transfer this
to semitopologies it splits into two notions — _weakly_ dense and _strongly_
dense (Definition 11.1.2) — both of which turn out to be important.

Sometimes, ideas that come from semitopology project down to topology but may
lose impact in doing so; they make mathematical sense, but become less
interesting, or at least lose finesse. For example: our theory of
semitopologies considers notions of _topen set_ and _strongly topen set_
(Definitions 3.2.1 and 3.7.5). In topology these are equivalent to one
another, and to a known and simpler topological property of being
_hyperconnected_ (Definition 3.7.12).757575…but (strong) topens are their own
thing. Analogy: a projection from $\mathbb{C}$ to $\mathbb{R}$ maps $a+bi$ to
$a$; this is not evidence that $i$ is equivalent to $0$! Something similar
happens with the semitopological notion of _strongly dense for_ ; see the
discussion in Remark 11.4.7.

  5. 5.

Semitopological questions such as _‘is this a topen set’_ or _‘are these two
points intertwined’_ or _‘does this point have a topen neighbourhood’_ — and
many other definitions in this paper, such as our taxonomy of points into
_regular_ , _weakly regular_ , _indirectly regular_ , _quasiregular_ ,
_unconflicted_ , and _hypertransitive_ ; or the notions of _witnessed set_ ,
and _kernel_ …and so on — appear to be novel.

Also in the background is that we are particularly interested in properties
and algorithms that work well using local and possibly incomplete or even
partially incorrect information.

Thus semitopologies have their own distinct character: because they are
mathematically distinct, and because modern applications having to do with
actionable coalitions and distributed systems motivate us to ask questions
that have not necessarily been considered before.

###  12.2 Related work

##### Semiframes

This paper is about point-set semitopologies: everything we do is in concrete
sets, just as in point-set topology.

Yet, a semitopology is just a complete subsemilattice of a powerset — just as
a topology is just a complete and finitely co-complete sublattice of a
powerset. The reader may know of _pointless topology_ , which abstracts away
the concrete powerset to a pointless (i.e. point-free) theory of _locales_ ;
an accessible and brief discussion is in [Joh83].

A companion paper [GL23] (unpublished draft, submitted for publication) treats
semitopologies from a pointless, algebraic, and categorical perspective. We
set up categories of semitopologies and semiframes and construct a categorical
duality between them, consciously imitating the classic duality between
topologies and frames — and this is informative, since the proofs require
semiframes to be more than just semilattices; they come out as something else.
We also give algebraic characterisations of the regularity properties
considered in this paper, such as those mentioned in Definition 4.1.4.

##### Union sets and minimal structures

There is a thread of research into _union-closed families_ ; these are subsets
of a finite powerset closed under unions, so that a union-closed families is
precisely just a finite semitopology. The motivation is to study the
combinatorics of finite subsemilattices of a powerset. Some progress has been
made in this [Poo92]; the canonical reference for the relevant combinatorial
conjectures is the ‘problem session’ on page 525 (conjectures 1.9, 1.9’, and
1.9”) of [Riv85]. See also recent progress in a conjecture about union-closed
families (permalink).

There is no direct connection to this work, though the combinatorial
properties considered may yet become useful for proving properties of concrete
algorithms.

A _minimal structure_ on a set $X$ is a subset of $\mathit{pow}(X)$ that
contains $\varnothing$ and $X$. Thus a semitopology is a minimal structure
that is also closed under arbitrary unions. There is a thread of research into
minimal structures, studying how notions familiar from topology (such as
continuity) fare in weak (minimal) settings [PN01] and how this changes as
axioms (such as closure under unions) are added or removed. An accessible
discussion is in [Szá07], and see the brief but comprehensive references in
Remark 3.7 of that paper. Of course our focus is on properties of
semitopologies which are not considered in that literature; but we share an
observation with minimal structures that it is useful to study topology-like
constructs, in the absence of closure under intersections.

##### Gradecast converges on a topen

Many consensus algorithms have the property that once consensus is established
in a quorum $O$, it propagates to $|O|$. For example, in the Grade-Cast
algorithm [FM88], participants assign a confidence grade of 0, 1 or 2 to their
output and must ensure that if any participant outputs $v$ with grade 2 then
all must output $v$ with grade at least 1. If all the quorums of a participant
intersect some set $S$ that unanimously supports value $v$, then the
participant assigns grade at least 1 to $v$.

From the view of our paper, this is just taking a closure, which suggests
that, to convince a topen to agree on a value, it would suffice to first
convince an open neighbourhood that intersects the topen, and then use Grade-
Cast to convince the whole topen. More on this in Proposition 5.3.2 and Remark
5.3.3.

##### Algebraic topology as applied to distributed computing tasks

Continuing the discussion of tasks above, the reader may know that solvability
results about distributed computing tasks have been obtained from algebraic
topology, starting with the impossibility of wait-free $k$-set consensus using
read-write registers and the Asynchronous Computability Theorem [HS93, BG93,
SZ93] in 1993. See [HKR13] for numerous such results.

The basic observation is that the set of final states of a distributed
algorithm forms a simplicial complex, called the _protocol complex_ , and
topological properties of this complex, like connectivity, are constrained by
the underlying communication and fault model. These topological properties in
turn can determine what tasks are solvable. For example: every algorithm in
the wait-free model with atomic read-write registers has a connected protocol
complex, and because the consensus task’s output complex is disconnected,
consensus in this model is not solvable [HKR13, Chapter 4].

This paper is also topological, but in a different way: we use
(semi)topologies to study consensus in and of itself, rather than the
solvability of consensus or other tasks in particular computation models. Put
another way: the papers cited above use topology to study the solvability of
distributed tasks, but this paper shows how the very idea of ‘distribution’
can be viewed as having a semitopological foundation.

Of course we can imagine that these might be combined — that in future work we
may find interesting and useful things to say about the topologies of
distributed algorithms when viewed as algorithms _on_ and _in_ a semitopology.

##### Fail-prone systems and quorum systems

Given a set of processes $\bm{\mathsf{P}}$, a _fail-prone_ system [MR98] (or
_adversary structure_ [HM00]) is a set of _fail-prone sets_
$\mathcal{F}=\\{F_{1},...,F_{n}\\}$ where, for every $1\leq i\leq n$,
$F_{i}\subseteq\bm{\mathsf{P}}$. $\mathcal{F}$ denotes the assumptions that
the set of processes that will fail (potentially maliciously) is a subset of
one of the fail-prone sets. A _dissemination quorum system_ for $\mathcal{F}$
is a set $\\{Q_{1},...,Q_{m}\\}$ of quorums where, for every $1\leq i\leq m$,
$Q_{i}\subseteq\bm{\mathsf{P}}$, and such that

for every two quorums $Q$ and $Q^{\prime}$ and for every fail-prone set $F$,
$\left(Q\cap Q^{\prime}\right)\setminus F\neq\emptyset$ and

for every fail-prone set $F$, there exists a quorum disjoint from $F$. Several
distributed algorithms, such as Bracha Broadcast [Bra87] and PBFT [CL02], rely
on a quorum system for a fail-prone system $\mathcal{F}$ in order to solve
problems such as reliable broadcast and consensus assuming (at least) that the
assumptions denoted by $\mathcal{F}$ are satisfied.

Several recent works generalise the fail-prone system model. Under the failure
assumptions of a traditional fail-prone system, Bezerra et al. [BKK22] study
reliable broadcast when participants each have their own set of quorums.
Asymmetric Fail-Prone Systems [CT19] generalise fail-prone systems to allow
participants to make different failure assumption and have different quorums.
In Permissionless Fail-Prone Systems [CLZ23], participants not only make
assumptions about failures, but also make assumptions about the assumptions of
other processes; the resulting structure seems closely related to witness
semitopologies, but the exact relationship still needs to be elucidated.

In Federated Byzantine Agreement Systems [Maz15], participants declare quorum
slices and quorums emerge out of the collective quorum slices of their
members. Quorum slices are a special case of the notion of witness-set in
Definition 8.2.2(8.2.2). García-Pérez and Gotsman [GPG18] rigorously prove the
correctness of broadcast abstractions in Stellar’s Federated Byzantine
Agreement model and investigate the model’s relationship to dissemination
quorum systems. The Personal Byzantine Quorum System model [LGM19] is an
abstraction of Stellar’s Federated Byzantine Agreement System model and
accounts for the existence of disjoint consensus clusters (in the terminology
of the paper) which can each stay in agreement internally but may disagree
between each other. Consensus clusters are closely related to the notion of
topen in Definition 3.2.1(3.2.1).

Sheff et al. study heterogeneous consensus in a model called Learner Graphs
[SWRM21] and propose a consensus algorithm called Heterogeneous Paxos.

Cobalt, the Stellar Consensus Protocol, Heterogeneous Paxos, and the Ripple
Consensus Algorithm [Mac18, Maz15, SWRM21, SYB14] are consensus algorithms
that rely on heterogeneous quorums or variants thereof. The Stellar network
[LLM+19] and the XRP Ledger [SYB14] are two global payment networks that use
heterogeneous quorums to achieve consensus among an open set of participants;
the Stellar network is an instance of a witness semitopology.

Quorum systems and semitopologies are not the same thing. Quorum systems are
typically taken to be such that all quorums intersect (in our terminology:
they are _intertwined_), whereas semitopologies do not require this. On the
other hand, quorums are not always taken to be closed under arbitrary unions,
whereas semitopologies are (see the discussion in Example 2.1.4(7)).

But there are also differences in how the maths has been used and understood.
This paper has been all about point-set topology flavoured ideas, whereas the
literature on fail-prone systems and quorum systems has been most interested
in synchronisation algorithms for distributed systems. We see these interests
as complementary, and the difference in emphasis is a feature, not a bug. Some
work by the second author and others [LGM19] gets as far as proving an
analogue to Lemma 3.5.2 (though we think it is fair to say that the
presentation in this paper is much simpler and more clear), but it fails to
notice the connection with topology and the subsequent results which we
present in this paper.

###  12.3 Future work

There is a great deal more to say than we have covered in this
paper.767676…for all its length, it’s just a start. We briefly outline the
many ways in which this work can be extended and improved:

  1. 1.

In Definition 2.1.3 we define a _value assignment_
$f:\bm{\mathsf{P}}\to\mathsf{Val}$ to be a function from a semitopology to a
codomain $\mathsf{Val}$ that is given the discrete semitopology. This is a
legitimate starting point, but of course we should consider more general
codomains. This could include an arbitrary semitopology on the right (for
greatest generality), but even for our intended special case of consensus it
would be interesting to try to endow $\mathsf{Val}$ with a semilattice
structure (or something like it), at least, e.g. to model merging of distinct
updates to a ledger.777777We write ‘something like it’ because we might also
consider, or consider excluding, possibly conflicting updates. We can easily
generate a (semi)topology from a semilattice by taking points to be elements
of the lattice and open sets to be up-closed sets, and this would be a natural
generalisation of the discrete semitopologies we have used so far.

  2. 2.

We have not considered morphisms of semitopologies and how to organise
semitopologies into a category. Since this paper was drafted, we have
completed a treatment of the category of semitopologies, along with a
categorical duality with a category of what we call _semiframes_ , and of
algebraic versions of the anti-separation and regularity properties considered
in this paper. This is described in [GL23] (submitted and under review).

  3. 3.

We have discussed dense subsets, but not (yet) considered notions of sequence
or net convergence. Nor have we considered what would correspond to the
exponential (or _Vietoris_) semitopology. Semilattice representation results
exist [Bre84], but a design space exists here and we should look for
representations well-suited to computationally verifying or refuting
properties of witness semitopologies.

  4. 4.

We mentioned in Subsection 12.2 that this paper on semitopology is not about
algebraic topology applied to solvability of distributed computing tasks.
These are distinct topics, and the fact that they share a word in their name
does not make them any more equal than a Great Dane and a Danish pastry.

But, it is a very interesting question what algebraic _semi_ topology might
look like. To put this another way: what is the geometry of semitopological
spaces? We would very much like to know.

  5. 5.

It remains to consider Byzantine behaviour, by which we mean that some
participants may misreport their view of the network in order to ‘invent’ or
sabotage quorums and so influence the outcome of consensus.

So for instance we can ask: “What conditions can we put on a witness function
to guarantee that changing the witness function at one point $p$ will not
change $\mathit{ker}(p^{\prime})$ for any $p^{\prime}\neq p$?”. Thus
intuitively, given a semitopology $(\bm{\mathsf{P}},\mathsf{Open})$ we are
interested in asking how properties range over an ‘$\epsilon$-ball’ of
perturbed semitopologies — as might be caused by various possible non-standard
behaviours from a limited number of Byzantine points — and in particular we
are looking for criteria to guarantee that appropriately-chosen good
properties be preserved under perturbation.

  6. 6.

This paper studies how consensus, once achieved on an open set $O$, propagates
to its closure $|O|$; see Proposition 5.3.2 and Remark 5.3.3. But this is just
half of the problem of consensus: it remains to consider (within our
semitopological framework) what it is to attain consensus on some open set in
the first place. That is: suppose $(\bm{\mathsf{P}},\mathsf{Open})$ is a
semitopology and $f:\bm{\mathsf{P}}\to\mathsf{Var}$ is a value assignment.
Then what does it mean, in maths and algorithms, to find a value assignment
$f^{\prime}:\bm{\mathsf{P}}\to\mathsf{Var}$ that is ‘close’ to $f$ but is
continuous on some open set $O$? In this paper we have constructed a detailed
theory of what it would then be to extend $f^{\prime}$ to an
$f^{\prime\prime}$ that continuously extends $f^{\prime}$ to regular points;
but we have not yet looked at how to build the $f^{\prime}$. We speculate that
unauthenticated Byzantine consensus algorithms (like Information-Theoretic
HotStuff [AS20]) can be understood in our setting; unlike authenticated
algorithms, unauthenticated algorithms do not rely on one participant being
able to prove to another, by exhibiting signed messages, that a quorum has
acted in a certain way.

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