# Separability Properties of Nilpotent $\mathbb{Q}[x]$-Powered Groups II

Stephen Majewicz  Department of Mathematics
CUNY-Kingsborough Community College
Brooklyn, New York 11235<EMAIL_ADDRESS>and  Marcos Zyman
Department of Mathematics
CUNY-Borough of Manhattan Community College
New York, New York 10007<EMAIL_ADDRESS>
(Date: December 20, 2023)

###### Abstract.

In this paper, we study nilpotent $\mathbb{Q}[x]$-powered groups that satisfy
the following property: For some set of primes $\omega$ in $\mathbb{Q}[x],$
every $\omega^{\prime}$-isolated $\mathbb{Q}[x]$-subgroup in some family of
its $\mathbb{Q}[x]$-subgroups is finite $\omega$-type separable.

###### Key words and phrases:

nilpotent group, nilpotent $R$-powered group, nilpotent
$\mathbb{Q}[x]$-powered group, subgroup separability

###### 2000 Mathematics Subject Classification:

Primary 20F18, 20F19, 20E26; Secondary 13C12, 13C13, 13G05

††copyright: ©2006: American Mathematical Society

##  1\. Introduction

This paper is a continuation of our paper entitled “Separability Properties of
Nilpotent $\mathbb{Q}[x]$-Powered Groups” [12].

Let $\mathcal{C}$ be a class of groups. A group $G$ is _residually
$\mathcal{C}$_ if for every $1\neq g\in G,$ there exists $K\in\mathcal{C}$ and
a homomorphism $\varphi$ from $G$ onto $K$ such that $\varphi(g)\neq 1.$ A
subgroup $H$ of a group $G$ is called _$\mathcal{C}$ -separable_ in $G$ if,
for every element $g\in G\setminus H,$ there is a homomorphism $\varphi$ of
$G$ onto $K\in\mathcal{C}$ such that $\varphi(g)\notin\varphi(H).$ It is
immediate that $G$ is residually $\mathcal{C}$ if and only if
$\left\\{1\right\\}$ is $\mathcal{C}$-separable in $G$.

Let $\mathcal{P}$ be a non-empty set of primes and $\mathcal{P}^{\prime}$ the
set of all primes not in $\mathcal{P}.$ A natural number $n$ is called a
_$\mathcal{P}$ -number_ if every prime divisor of $n$ belongs to
$\mathcal{P}.$ By convention, $1$ is a $\mathcal{P}$-number for any set of
primes $\mathcal{P}.$ A $\mathcal{P}$-group is a torsion group such that each
of its elements has order a $\mathcal{P}$-number. Denote the class of all
finite $\mathcal{P}$-groups by $\mathcal{F}_{\mathcal{P}},$ and let $G$ be any
group with $H\leq G.$ If $H$ is $\mathcal{F}_{\mathcal{P}}$-separable in $G,$
then it can be shown that $H$ is a $\mathcal{P}^{\prime}$-isolated subgroup of
$G$ (i.e. if $h\in G$ and $h^{r}\in H$ for some $\mathcal{P}^{\prime}$-number
$r,$ then $h\in H$).

The converse, however, fails in general. For instance, in [1], Bardakov showed
that if $G$ is a non-abelian group and $H$ is an isolated (finitely generated)
subgroup of $G,$ then $H$ is not separable in the class of nilpotent groups
(hence, $H$ is not $\mathcal{F}_{\\{p\\}}$-separable in $G$ for all primes
$p$).

In [13], Sokolov continued the study of
$\mathcal{F}_{\mathcal{P}}$-separability of subgroups in nilpotent groups. A
group $G$ is said to have _property $\mathcal{G}_{\mathcal{P}}$_ (for some
family of its subgroups) if every $\mathcal{P}^{\prime}$-isolated subgroup of
$G$ belonging to the family is $\mathcal{F}_{\mathcal{P}}$-separable. Prior to
Sokolov’s work, it was shown by Loginova [6] that for every prime $p,$ all
finitely generated nilpotent groups have property $\mathcal{G}_{\\{p\\}}.$
This result can be generalized for a set of primes rather than a single prime.
Two of Sokolov’s results are:

  1. (1)

An abelian group $A$ has property $\mathcal{G}_{\mathcal{P}}$ if and only if
$A$ satisfies the following condition: In every quotient group $B$ of $A,$ all
$p$-primary components of the torsion subgroup of $B$ have finite exponent for
all $p\in\mathcal{P}.$

  2. (2)

Every $\mathcal{P}$-restricted nilpotent group has property
$\mathcal{G}_{\mathcal{P}}.$

(An abelian group is _$\mathcal{P}$ -restricted_ if in every quotient $B$ of
$A,$ all $p$-primary components of the torsion subgroup of $B$ are finite for
all $p\in\mathcal{P}.$ A nilpotent group is _$\mathcal{P}$ -restricted_ if it
has a finite central series with $\mathcal{P}$-restricted abelian factors.)

  3. (3)

A torsion-free nilpotent group has property $\mathcal{G}_{\mathcal{P}}$ if and
only if it is $\mathcal{P}$-restricted.

In this paper, we prove analogues of these results for nilpotent
$\mathbb{Q}[x]$-powered groups. The ring $\mathbb{Q}[x]$ is an example of a
_binomial ring_ , an integral domain with unity $R$ that has characteristic
zero such that for any $r\in R$ and positive integer $k,$

$\binom{r}{k}=\frac{r(r-1)\cdots(r-k+1)}{k!}\in R.$

A _nilpotent $R$-powered group_ is a nilpotent group $G$ that comes equipped
with an action by $R$ such that $g^{\alpha}$ is an element of $G$ for all
$g\in G$ and $\alpha\in R$, and satisfies the following:

  1. (1)

$g^{1}=g,\,g^{\alpha}g^{\beta}=g^{\alpha+\beta},$ and
$(g^{\alpha})^{\beta}=g^{\alpha\beta}$ for all $g\in G$ and $\alpha,\beta\in
R.$

  2. (2)

$(h^{-1}gh)^{\alpha}=h^{-1}g^{\alpha}h$ for all $g,\ h\in G$ and for all
$\alpha\in R.$

  3. (3)

If $\\{g_{1},\,\ldots,\,g_{n}\\}\subset G$ and $\alpha\in R,$ then

$g_{1}^{\alpha}\cdots
g_{n}^{\alpha}=\tau_{1}(\bar{g})^{\alpha}\tau_{2}(\bar{g})^{\binom{\alpha}{2}}\cdots\tau_{k-1}(\bar{g})^{\binom{\alpha}{k-1}}\tau_{k}(\bar{g})^{\binom{\alpha}{k}},$

where $k$ is the nilpotency class of the group generated by
$\\{g_{1},\,g_{2},\,\ldots,\,g_{n}\\},$ $\bar{g}=(g_{1},\,\ldots,\,g_{n}),$
and $\tau_{i}(\bar{g})$ is the $i$th _Hall-Petresco word_.

An _abelian $R$-group_ is a nilpotent $R$-powered group of class $1.$ If $G$
is an abelian $R$-group and $g,\,h\in G,$ then
$(gh)^{\alpha}=g^{\alpha}h^{\alpha}$ for any $\alpha\in R.$ This follows from
(3), together with the fact that $\tau_{1}(g,\,h)=gh.$ Thus, every abelian
$R$-group can be viewed as an $R$-module. Throughout the paper, we use
multiplicative and exponential notation for the operations in an abelian
$R$-group.

Nilpotent $R$-powered groups naturally arise from taking the $R$-completion of
a finitely generated torsion-free nilpotent group with respect to a given
Mal’cev basis. Notions such as $R$-subgroup, $R$-morphism, direct products,
$R$-series, etc. are defined in the obvious way (see [4], [7], and [14]).

In this paper, all direct products will be restricted.

All necessary background material will be provided in the preliminary section.
Throughout the paper, $R$ will always be a binomial ring (possibly with other
qualifications). Unless otherwise told, we use “$1$” to denote the unity
element in a ring and the identity element in a group.

The definitions of residually $\mathcal{C}$ and $\mathcal{C}$-separable carry
over to the category of nilpotent $R$-powered groups in a natural way.

######  Definition 1.1.

Let $\mathcal{C}$ be a class of nilpotent $R$-powered groups, and let $G$ be
any nilpotent $R$-powered group.

  * •

$G$ is _residually $\mathcal{C}$_ if for every $1\neq g\in G,$ there exists
$K\in\mathcal{C}$ and an $R$-homomorphism $\varphi$ from $G$ onto $K$ such
that $\varphi(g)\neq 1.$ Equivalently, there exists a normal $R$-subgroup
$N_{g}$ of $G$ with $g\notin N_{g}$ such that $G/N_{g}\in\mathcal{C}.$

  * •

An $R$-subgroup $H$ of $G$ is called _$\mathcal{C}$ -separable_ in $G$ if, for
every element $g\in G\setminus H,$ there is an $R$-homomorphism $\varphi$ of
$G$ onto $K\in\mathcal{C}$ such that $\varphi(g)\notin\varphi(H).$
Equivalently, for every element $g\in G\setminus H,$ there exists a normal
$R$-subgroup $N$ of $G$ such that $G/N\in\mathcal{C}$ and $g\notin HN.$

We introduce some notation. Let $R$ be a UFD. We will assume throughout the
paper that all sets of primes in $R$ are non-empty and no two elements in a
given set are associates. If $\omega$ is such a set, then $\omega^{\prime}$
will denote any set of primes satisfying the following:

  * •

No element in $\omega$ has an associate in $\omega^{\prime}.$

  * •

Every prime in $R$ has exactly one associate in either $\omega$ or
$\omega^{\prime}.$

For a set of primes $\omega,$ we denote the class of nilpotent $R$-powered
groups of finite $\omega$-type by $\mathcal{FT_{\omega}}$ (see Definition 2.4
below). The following definitions mimic those in Sokolov’s paper:

  * •

A nilpotent $R$-powered group $G$ has _property $\mathcal{G}_{\omega}$_ for
some family of its $R$-subgroups if every $\omega^{\prime}$-isolated
$R$-subgroup in this family is $\mathcal{FT_{\omega}}$-separable.

  * •

An abelian $R$-group $G$ satisfies _Condition ( $B_{\omega}$)_ if for every
$R$-quotient $G/N$ of $G$ and every $\pi\in\omega,$ the $\pi$-primary
component of $G/N$ is bounded.

  * •

An abelian $R$-group $G$ is called _$\omega$ -restricted_ if for every
$R$-quotient $G/N$ of $G$ and every $\pi\in\omega,$ the $\pi$-primary
component of $G/N$ is finitely $R$-generated (and, thus, finite
$\omega$-type).

  * •

A nilpotent $R$-powered group is _$\omega$ -restricted_ if it has a finite
central $R$-series with $\omega$-restricted abelian $R$-factors.

Our focus will be on the case $R=\mathbb{Q}[x].$ Let $\omega$ be a set of
primes in $\mathbb{Q}[x].$ The main results in this paper are:

  * •

A necessary and sufficient condition for an abelian $\mathbb{Q}[x]$-group to
have property $\mathcal{G}_{\omega}$ is that is satisfies Condition
($B_{\omega}$).

  * •

Every $\omega$-restricted nilpotent $\mathbb{Q}[x]$-powered group has property
$\mathcal{G}_{\omega}$ for its family of normal $\mathbb{Q}[x]$-subgroups.

  * •

If $G$ is a $\mathbb{Q}[x]$-torsion-free nilpotent $\mathbb{Q}[x]$-powered
group and has property $\mathcal{G}_{\omega},$ then each
$\mathbb{Q}[x]$-factor group $\zeta_{i+1}G/\zeta_{i}G$ of its upper central
$\mathbb{Q}[x]$-series has property $\mathcal{G}_{\omega}.$

##  2\. Preliminaries

In this section, we provide the terminology and results on nilpotent
$R$-powered groups which will be needed in this paper. Let $G$ be a nilpotent
$R$-powered group. The following appear in [3], [4], and [14]:

  * •

If $H$ is an $R$-subgroup (normal $R$-subgroup) of $G,$ then we write
$H\leq_{R}G$ ($H\unlhd_{R}G$). We say that $H$ is _$R$ -generated by
$X\subseteq G$_ if $H$ is the smallest $R$-subgroup of $G$ containing $X.$ In
this situation, we write $H=gp_{R}(X).$ If $X$ is finite, then we say that $H$
is _finitely $R$-generated_.

Notation: If $\alpha\in R,$ then $G^{\alpha}=gp_{R}(g^{\alpha}\ |\ g\in G).$
Note that $G^{\alpha}\unlhd_{R}G.$

  * •

If $N\unlhd_{R}G,$ then the $R$-action on $G$ induces an $R$-action on $G/N,$

$(gN)^{\alpha}=g^{\alpha}N\hbox{ for all }gN\in G/N\hbox{ and }\alpha\in R,$

which turns $G/N$ into a nilpotent $R$-powered group.

  * •

If $H\leq_{R}G$ and $N\unlhd_{R}G,$ then $HN\leq_{R}G$ and $HN=gp_{R}(H,\ N),$
the smallest $R$-subgroup containing $H$ and $N$. Furthermore, the kernel of
an $R$-homomorphism $\varphi:G_{1}\rightarrow G_{2}$ is a normal $R$-subgroup
of $G_{1}.$ It follows that the usual isomorphism theorems carry over to
nilpotent $R$-powered groups. We will refer to them as the _$R$ -isomorphism
theorems_. If $G_{1}$ and $G_{2}$ are $R$-isomorphic nilpotent $R$-powered
groups, then we write $G_{1}\cong_{R}G_{2}.$

  * •

All upper central subgroups of $G,$ written as $\zeta_{i}G,$ are $R$-subgroups
of $G.$ The center of $G$ will be denoted by $Z(G).$

The concept of a $P$-number for a set of primes $P\subset\mathbb{N}$ is
generalized in the next definition which is a slight improvement of our
earlier definition in [11]. We remind the reader that all sets of primes in
$R$ are non-empty and no pair of elements in such sets are associates.

######  Definition 2.1.

Let $R$ be a UFD and $\omega$ a set of primes in $R.$ A non-zero element
$\alpha\in R$ is an _$\omega$ -member_ if $\alpha$ is a non-unit and each
prime divisor of $\alpha$ has an associate in $\omega.$

######  Definition 2.2.

[11] Let $R$ be a UFD and $\omega$ a set of primes in $R.$ Suppose that $G$ is
a nilpotent $R$-powered group. An element $g\in G$ is an _$\omega$ -torsion
element_ if $g^{\alpha}=1$ for some $\omega$-member $\alpha.$ The set of
$\omega$-torsion elements of $G$ is written as $\tau_{\omega}(G).$ If every
element of $G$ is $\omega$-torsion, then $G$ is an _$\omega$ -torsion group_.
We say that $G$ is _$\omega$ -torsion-free_ if the only $\omega$-torsion
element of $G$ is $1.$

If $\pi$ is a prime in $R$ and $\omega=\\{\pi\\},$ then we use the terms
$\pi$-torsion and $\pi$-torsion-free and write $\tau_{\pi}(G)$ for the set of
$\pi$-torsion elements of $G.$ It is clear that if $\omega_{1}$ and
$\omega_{2}$ are sets of primes in $R,$ then
$\tau_{\omega_{1}}(G)=\tau_{\omega_{2}}(G)$ if every element in $\omega_{1}$
has an associate in $\omega_{2}$ and vice versa. In particular,
$\tau_{\pi_{1}}(G)=\tau_{\pi_{2}}(G)$ whenever $\pi_{1}$ and $\pi_{2}$ are
prime associates.

The next definition does not require $R$ to be a UFD.

######  Definition 2.3.

[14] Let $G$ be a nilpotent $R$-powered group.

  * •

An element $g\in G$ is an _$R$ -torsion element_ if $g^{\alpha}=1$ for some
$0\neq\alpha\in R.$ The set of $R$-torsion elements of $G$ is denoted by
$\tau(G).$

  * •

If every element of $G$ is $R$-torsion, then $G$ is called an _$R$ -torsion
group_. We say that $G$ is _$R$ -torsion-free_ if the only $R$-torsion element
of $G$ is $1.$

######  Theorem 2.1.

[11], [14] Let $G$ be a nilpotent $R$-powered group.

  * •

If $R$ is a UFD and $\omega$ is a set of primes in $R,$ then
$\tau_{\omega}(G)\unlhd_{R}G$ and $G/\tau_{\omega}(G)$ is $\omega$-torsion-
free.

  * •

If $R$ is any binomial ring, then $\tau(G)\unlhd_{R}G$ and $G/\tau(G)$ is
$R$-torsion-free.

######  Definition 2.4.

[11] Let $R$ be a UFD and $\omega$ a set of primes in $R.$ A finitely
$R$-generated $\omega$-torsion group $G$ is said to be of _finite
$\omega$-type_. If $\omega=\\{\pi\\}$ for a prime $\pi\in R,$ then $G$ is of
_finite $\pi$-type_.

The classes of nilpotent $R$-powered groups of finite $\omega$-type and finite
$\pi$-type will be denoted by $\mathcal{FT_{\omega}}$ and $\mathcal{FT_{\pi}}$
respectively.

The next definition does not require $R$ to be a UFD.

######  Definition 2.5.

[7] A nilpotent $R$-powered group is of _finite type_ if it is a finitely
$R$-generated $R$-torsion group.

The class of nilpotent $R$-powered groups of finite type will be denoted by
$\mathcal{FT}.$

######  Theorem 2.2.

[5] If $R$ is a noetherian domain, then every $R$-subgroup of a finitely
$R$-generated nilpotent $R$-powered group is finitely $R$-generated.

######  Corollary 2.3.

[8] Let $G$ be a nilpotent $R$-powered group, where $R$ is a noetherian
domain, and $\omega$ a set of primes in $R.$

  * •

If $G\in\mathcal{FT}$ and $H\leq_{R}G,$ then $H\in\mathcal{FT}.$ If
$H\unlhd_{R}G,$ then $G/H\in\mathcal{FT}.$

  * •

Suppose that $R$ is also a UFD. If $G\in\mathcal{FT_{\omega}}$ and
$H\leq_{R}G,$ then $H\in\mathcal{FT_{\omega}}.$ If $H\unlhd_{R}G,$ then
$G/H\in\mathcal{FT_{\omega}}.$

It is well-known that the torsion subgroup of a nilpotent group is a direct
product of its $p$-torsion subgroups, where $p$ ranges over the primes (see
[3]). Similarly, we have:

######  Theorem 2.4.

[7] Let $R$ be a PID and $G$ a nilpotent $R$-powered group. Let $\mathbb{P}$
be a set of primes in $R$ such that each prime in $R$ has exactly one
associate in $\mathbb{P},$ and let $\omega$ be a set of primes in $R.$

  * •

If $G$ is an $R$-torsion group, then
$G=\prod_{\pi\in\mathbb{P}}\tau_{\pi}(G).$

  * •

If $G$ is an $\omega$-torsion group, then
$G=\prod_{\pi\in\omega}\tau_{\pi}(G).$

Next we define the order ideal of an element belonging to a nilpotent
$R$-powered group and the exponent ideal of a nilpotent $R$-powered group.
Some of this appears in [8] and mimics the definitions from module theory.

######  Definition 2.6.

Let $R$ be a PID and $G$ a nilpotent $R$-powered group with $g\in G.$ The set

$Ann(g)=\\{\alpha\in R\ |\ g^{\alpha}=1\\}$

is an ideal of $R$ called the _order ideal_ or _annihilator_ of $g.$ We say
that $g$ has _infinite (or unbounded) order_ whenever $Ann(g)=\\{0\\}.$

Suppose that $Ann(g)\neq\\{0\\}.$ Since $R$ is a PID, there exists an element
$0\neq\beta\in R$ such that $Ann(g)=\,<\beta>,$ where $<\beta>$ is the ideal
generated by $\beta.$

######  Definition 2.7.

The element $\beta$ is called the _non-infinite (or bounded) order_ of $g.$

Clearly, any associate of $\beta$ also generates $Ann(g).$ Hence, the order of
$g$ is unique up to a unit factor.

######  Definition 2.8.

Let $R$ be a PID. The _exponent ideal_ of a nilpotent $R$-powered group $G$ is

$Ann(G)=\\{\alpha\in R\ |\ g^{\alpha}=1\hbox{ for all }g\in G\\}.$

If $Ann(G)=\\{0\\},$ then $G$ has _infinite (or unbounded) exponent_. In this
case, $G$ is said to be _unbounded_. Otherwise, $G$ is termed _bounded_.

As before, there exists an element $0\neq\lambda\in R$ such that
$Ann(G)=\,<\lambda>$ whenever $Ann(G)\neq\\{0\\}.$

######  Definition 2.9.

The element $\lambda$ is the _non-infinite (or bounded) exponent_ of $G.$

The bounded exponent $\lambda$ of $G$ is unique up to a unit factor since any
associate of $\lambda$ also generates $Ann(G).$

Every bounded nilpotent $R$-powered group is an $R$-torsion group. However, an
$R$-torsion group is not necessarily bounded. Furthermore, if $G$ is a
nilpotent $R$-powered group and $\alpha\in R,$ then $G/G^{\alpha}$ is bounded.

Examples: Let $R$ be a PID.

1) Let $\omega$ be a set of primes in $R.$ Each element of a nilpotent
$R$-powered group of finite $\omega$-type has bounded order and the group is
bounded (see [8]). The same is true for nilpotent $R$-powered groups of finite
type.

2) Let $G$ be a nilpotent $R$-powered group and $\pi$ a prime in $R.$

  * •

The direct product

$G/G^{\pi}\times G/G^{\pi^{2}}\times G/G^{\pi^{3}}\times\cdots$

is an unbounded $\pi$-torsion group.

  * •

The direct product

$G/G^{\pi}\times G/G^{\pi}\times G/G^{\pi}\times\cdots$

is a bounded $\pi$-torsion group.

3) Let $\pi$ be a prime in $R.$ Suppose that $G$ is an abelian $R$-group with
$R$-generating set $\\{g_{1},\ g_{2},\ g_{3},\ \ldots\\}$ with $g_{1}\neq 1$
and subject to the relations

$g_{1}^{\pi}=1,\ g_{2}^{\pi}=g_{1},\ g_{3}^{\pi}=g_{2},\ \ldots.$

Then $G$ is an unbounded $\pi$-torsion group called the _$\pi$ -quasicyclic
$R$-group_. It will be denoted by $Q(\pi^{\infty}).$

The next theorem is well-known for modules (see [2]).

######  Theorem 2.5.

Let $R$ be a PID. Every bounded abelian $R$-group is a direct product of
cyclic $R$-groups.

An important result due to N. Blackburn states that if $g$ and $h$ are
elements in a nilpotent group and $p$ is a prime, then for any
$n\in\mathbb{N}$ the product $g^{p^{n}}h^{p^{n}}$ is a $p^{m}$th power of some
element of the group and some $m$ (see [3] or [14] for details). A related
result is:

######  Theorem 2.6.

[9] Suppose $R$ contains $\mathbb{Q},$ and let $G$ be a nilpotent $R$-powered
group. If $\beta\in R$ and $g_{1},\ \ldots,\ g_{m}\in G,$ then there exists
$h\in G$ such that

$g_{1}^{\beta}\cdots g_{m}^{\beta}=h^{\beta}.$

Hence, every element in $G^{\beta}$ is a $\beta$th power of an element of $G.$

######  Definition 2.10.

Let $G$ be an abelian $R$-group.

  * •

$G$ is called _$R$ -divisible_ if for every $g\in G$ and $0\neq\alpha\in R,$
there exists $h\in G$ such that $g=h^{\alpha}.$ We say that $h$ is an
_$\alpha$ th root_ of $g.$

  * •

If $0\neq\beta\in R,$ then $G$ is _$\beta$ -divisible_ if every element of $G$
has a $\beta$th root.

Any $R$-quotient of an $R$-divisible group is $R$-divisible whenever $R$ is a
domain. Furthermore, the direct product of a collection of $R$-divisible
($\beta$-divisible) groups is $R$-divisible ($\beta$-divisible).

It is easy to show that the $p$-quasicyclic group is divisible when $p$ is an
ordinary prime (see Lemma 5.25 in [3]). The same proof gives:

######  Lemma 2.7.

Let $R$ be a PID and $\pi$ a prime in $R$. The $\pi$-quasicyclic $R$-group is
$R$-divisible.

The following theorem is fundamental in the theory of modules.

######  Theorem 2.8.

[2] Every abelian $R$-group can be embedded in an $R$-divisible group.

The next few results involve abelian $R$-groups that are residually
$\mathcal{FT}.$

######  Lemma 2.9.

Let $R$ be a PID. Every cyclic $R$-group is residually $\mathcal{FT}.$

###### Proof.

Suppose that $G=gp_{R}(g)$ is a cyclic $R$-group. There are 2 cases to
consider.

Case 1: Assume $g$ has infinite order. Choose $\alpha,\ \beta\in R$ such that
$g^{\alpha}\notin gp_{R}\left(g^{\beta}\right),$ and set
$N=gp_{R}\left(g^{\beta}\right).$ If $g^{\lambda}\in G$ for some $\lambda\in
R,$ then $g^{\lambda}N\in G/N$ is an $R$-torsion element since

$\left(g^{\lambda}N\right)^{\beta}=\left(g^{\beta}N\right)^{\lambda}=N.$

And so, $G/N\in\mathcal{FT}.$ Furthermore, $g^{\alpha}$ is not in the kernel
of the natural map $G\rightarrow G/N.$ Therefore, $G$ is residually
$\mathcal{FT}.$

Case 2: Suppose that $g$ has bounded order and choose $0\neq\mu\in R$ such
that $Ann(g)=\,<\mu>.$ Since $G\cong_{R}R/Ann(g),$ we have that

$G\cong_{R}\frac{gp_{R}(g)}{gp_{R}\left(g^{\mu}\right)}.$

Clearly, $G\in\mathcal{FT}.$ ∎

If $\omega$ is a set of primes in a PID $R,$ then $\omega$-torsion cyclic
$R$-groups are clearly residually $\mathcal{FT_{\omega}}.$ Furthermore, if
$\mathcal{Q}$ is a property of nilpotent $R$-powered groups for any $R$ and
$G_{1},G_{2},\ \ldots$ are residually $\mathcal{Q},$ then the direct product
of the $G_{i}$ is also residually $\mathcal{Q}.$ This, together with Lemma
2.9, gives:

######  Corollary 2.10.

Let $R$ be a PID.

  1. (1)

A direct product of cyclic $R$-groups is residually $\mathcal{FT}.$

  2. (2)

If $\omega$ is a set of primes in $R,$ then a direct product of
$\omega$-torsion cyclic $R$-groups is residually $\mathcal{FT_{\omega}}.$

######  Theorem 2.11.

Let $R$ be a PID. An abelian $R$-group $G$ is residually $\mathcal{FT}$ if and
only if it does not contain non-trivial elements with $\beta$th roots for all
$\beta\in R.$

###### Proof.

Suppose that $g\in G$ ($g\neq 1$) has no $\alpha$th root for some $\alpha\in
R,$ so that $g\notin G^{\alpha}.$ Clearly, $G/G^{\alpha}$ is bounded. By
Theorem 2.5, $G/G^{\alpha}$ is a direct product of cyclic $R$-subgroups.
Hence, $G/G^{\alpha}$ is residually $\mathcal{FT}$ by Corollary 2.10.
Therefore, there exists a normal $R$-subgroup $N/G^{\alpha}$ of $G/G^{\alpha}$
such that $(G/G^{\alpha})/(N/G^{\alpha})$ is finite type and $g\notin N.$ And
so, $G$ is residually $\mathcal{FT}.$ The converse follows immediately from
the next lemma. ∎

######  Lemma 2.12.

If $G$ is a non-trivial $R$-divisible group and $H\in\mathcal{FT},$ then the
only $R$-homomorphism from $G$ to $H$ is the trivial one.

###### Proof.

Since $H$ is finite type, it has bounded exponent. Let $\lambda$ be the
exponent of $H,$ and let $g\in G.$ Since $G$ is $R$-divisible, there exists
$h\in G$ such that $g=h^{\lambda}.$ If $\varphi:G\rightarrow H$ is an
$R$-homomorphism, then

$\varphi(g)=\varphi\left(h^{\lambda}\right)=(\varphi(h))^{\lambda}=1.$

And so, $\varphi$ must be the trivial $R$-homomorphism. ∎

######  Definition 2.11.

[10] Let $R$ be a UFD and $\omega$ a set of primes in $R.$ An $R$-subgroup $H$
of a nilpotent $R$-powered group $G$ is _$\omega$ -isolated_ in $G$ if $g\in
G$ and $g^{\lambda}\in H$ for some $\omega$-member $\lambda$ imply that $g\in
H.$

“$R$-isolated” is defined in the obvious way and does not require a UFD. The
next result is obvious.

######  Lemma 2.13.

Let $R$ be a UFD and $G$ a nilpotent $R$-powered group. If $H\unlhd_{R}G,$
then $H$ is $\omega$-isolated in $G$ if and only if $G/H$ is $\omega$-torsion-
free.

Yet another concept that we will need is the height of a group element.

######  Definition 2.12.

Let $R$ be a PID and $\pi$ a prime in $R.$ Let $G$ be a $\pi$-torsion abelian
$R$-group. An element $g\in G$ has _height_ $n$ if $g$ has a $\pi^{n}$th root
in $G$ but no $\pi^{n+1}$th root in $G.$ If $g$ has a $\pi^{k}$th root for all
$k\in\mathbb{N},$ then $g$ is said to have _infinite height_.

We need to know when an abelian $R$-group can be expressed as a direct product
of cyclic $R$-groups. The next theorem provides two instances of when this can
be done. It is a generalization of well-known results due to Prüfer and
Kulikov.

######  Theorem 2.14.

[2] Let $R$ be a PID and $\pi$ a prime in $R.$

  1. (1)

A $\pi$-torsion abelian $R$-group $G$ is a direct product of cyclic
$R$-subgroups if and only if $G$ is the union of an ascending sequence

$G_{1}\subseteq G_{2}\subseteq\cdots\subseteq G_{n}\subseteq\cdots$

of $R$-subgroups, such that the heights in $G$ of the non-identity elements in
$G_{n}$ are bounded.

  2. (2)

A countably $R$-generated $\pi$-torsion abelian $R$-group without elements of
infinite height is a direct product of cyclic $R$-subgroups.

We end this section with a structure theorem for certain $R$-divisible groups.

######  Theorem 2.15.

[2] Let $R$ be a PID. Every $R$-divisible group is $R$-isomorphic to a direct
product of copies of $Q(\pi^{\infty})$ for various primes $\pi\in R$ and
copies of the field of fractions of $R.$

##  3\. MAIN SECTION

Let $\mathcal{C}$ be a class of nilpotent $R$-powered groups. Two
relationships between $\mathcal{C}$-separability and $\mathcal{C}$-residual
properties for nilpotent $R$-powered groups are captured in the next lemma.
These are mentioned in [13] for ordinary nilpotent groups. We include the
proof for completeness.

######  Lemma 3.1.

Let $G$ be a nilpotent $R$-powered group.

  1. (1)

$G$ is residually $\mathcal{C}$ if and only if $\\{1\\}$ is
$\mathcal{C}$-separable in $G.$

  2. (2)

Let $H\unlhd_{R}G.$ If $R$-homomorphic images of groups in $\mathcal{C}$ are
in $\mathcal{C},$ then $G/H$ is residually $\mathcal{C}$ if and only if $H$ is
$\mathcal{C}$-separable in $G.$

###### Proof.

We only prove (2) since (1) is clear. Assume first that $G/H$ is residually
$\mathcal{C}.$ Let $g\in G\setminus H$, so that $gH\neq H$ in $G/H$. Then
there exists $K\in\mathcal{C}$ and an onto $R$-homomorphism
$\widehat{\varphi}:G/H\rightarrow K$ such that $\widehat{\varphi}(gH)\neq 1$
in $K$. Let $p:G\rightarrow G/H$ be the canonical $R$-homomorphism. Then
$\varphi=\widehat{\varphi}\circ p$ is an $R$-homomorphism from $G$ onto $K$.
By construction,

$\varphi(g)=\widehat{\varphi}(gH)\neq 1$

in $K$. If $\varphi(g)=\varphi(h)$ for some $h\in H$ we would have
$\varphi(g)=1$, which is impossible. This shows that
$\varphi(g)\notin\varphi(H)$, and hence, $H$ is $\mathcal{C}$-separable in
$G$.

For the converse, assume that $H$ is $\mathcal{C}$-separable in $G$ and let
$xH\neq H$ in $G/H$. Then $x\notin H$ and there exists
$\widehat{K}\in\mathcal{C}$ and an onto $R$-homomorphism

$\widehat{\varphi}:G\rightarrow\widehat{K}$

such that $\widehat{\varphi}(x)\notin\widehat{\varphi}(H)$. Consider the
nilpotent $R$-powered group $K=\widehat{K}/\widehat{\varphi}(H)$. Let

$\varphi:G/H\rightarrow K$

be given by $\varphi(gH)=\widehat{\varphi}(g)\widehat{\varphi}(H)\in K$. A
routine check confirms that $\varphi$ is a well-defined onto $R$-homomorphism.
Since $\mathcal{C}$ is closed under $R$-homomorphic images, $K\in\mathcal{C}$.
Finally, since $\widehat{\varphi}(x)\notin\widehat{\varphi}(H)$, we conclude
that $\varphi(xH)=\widehat{\varphi}(x)\widehat{\varphi}(H)\neq 1$. This proves
that $G/H$ is residually $\mathcal{C}$, as promised. ∎

For the rest of the paper, we restrict ourselves to the binomial ring
$\mathbb{Q}[x].$

Every prime in $\mathbb{Q}[x]$ has an associate that is a monic prime.
Therefore, without loss of generality, we work with monic prime polynomials.
Throughout the remainder of the paper, $\mathbb{M}$ will denote the set of all
monic primes in $\mathbb{Q}[x].$ Our assumptions for subsets of primes given
in the introduction can be stated thus:

  * •

Every set of primes is a non-empty subset of $\mathbb{M}.$

  * •

If $\omega$ is a set of primes, then every $\omega$-member is monic and
$\omega^{\prime}$ is the set of monic primes such that
$\omega\cap\omega^{\prime}=\emptyset$ and
$\omega\cup\omega^{\prime}=\mathbb{M}.$

######  Lemma 3.2.

Let $G$ be a nilpotent $\mathbb{Q}[x]$-powered group. If
$H\leq_{\mathbb{Q}[x]}G$ and $H$ is $\mathcal{FT_{\omega}}$-separable in $G$
for some set of primes $\omega$ in $\mathbb{Q}[x],$ then $H$ is
$\omega^{\prime}$-isolated in $G.$

###### Proof.

Suppose that $H$ is not $\omega^{\prime}$-isolated. Then there exists $g\in
G\setminus H$ and $\alpha$ an $\omega^{\prime}$-member such that
$g^{\alpha}\in H.$ By hypothesis, there is a nilpotent $\mathbb{Q}[x]$-powered
group of finite $\omega$-type $K$ and an $\mathbb{Q}[x]$-homomorphism
$\varphi$ from $G$ onto $K$ such that $\varphi(g)\notin\varphi(H).$ Since $K$
is $\omega$-torsion, there exists an $\omega$-member $\mu$ such that
$(\varphi(g))^{\mu}=\varphi(g^{\mu})=1.$ Now, $\alpha$ and $\mu$ are
relatively prime because $\alpha$ is not an $\omega$-member. Since
$\mathbb{Q}[x]$ is a PID, there exist $r,\,s\in\mathbb{Q}[x]$ such that
$r\alpha+s\mu=1.$ Noting that $g^{\alpha}\in H$ implies that $g^{r\alpha}\in
H,$ we get

$\varphi(g)=(\varphi(g))^{r\alpha+s\mu}=\varphi(g^{r\alpha})(\varphi(g^{\mu}))^{s}=\varphi(g^{r\alpha})\in\varphi(H),$

a contradiction. We conclude that $H$ is $\omega^{\prime}$-isolated. ∎

We would like to find conditions for nilpotent $\mathbb{Q}[x]$-powered groups
to satisfy the converse of Lemma 3.2. This motivates the next definition.

######  Definition 3.1.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ A nilpotent
$\mathbb{Q}[x]$-powered group $G$ has _property $\mathcal{G}_{\omega}$_ for
some family of its $\mathbb{Q}[x]$-subgroups if every
$\omega^{\prime}$-isolated $\mathbb{Q}[x]$-subgroup in this family is
$\mathcal{FT_{\omega}}$-separable.

For simplicity, we will just write “$G$ has property $\mathcal{G}_{\omega}$”
when the family of $\mathbb{Q}[x]$-subgroups of $G$ being referred to is the
family of all of its $\mathbb{Q}[x]$-subgroups.

We will be interested in those abelian $\mathbb{Q}[x]$-groups that satisfy the
following:

Condition ($B_{\omega}$): If $\omega$ is a set of primes in $\mathbb{Q}[x]$
and $G$ is an abelian $\mathbb{Q}[x]$-group, then for every
$\mathbb{Q}[x]$-quotient $G/N$ of $G$ and every $\pi\in\omega,$ the
$\pi$-primary component $\tau_{\pi}(G/N)$ of $G/N$ is bounded.

######  Proposition 3.3.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ An abelian
$\mathbb{Q}[x]$-group $G$ satisfies Condition ($B_{\omega}$) if and only if
none of the $\mathbb{Q}[x]$-quotients of $G$ contain $\pi$-quasicyclic
$\mathbb{Q}[x]$-subgroups for every $\pi\in\omega.$

###### Proof.

Assume that there is a factor $\mathbb{Q}[x]$-group $G/N$ and a prime
$\pi\in\omega$ such that $\tau_{\pi}(G/N)$ is unbounded. We assert that some
$\mathbb{Q}[x]$-quotient of $G$ has a $\pi$-quasicyclic
$\mathbb{Q}[x]$-subgroup.

Let $k_{i}N$ be a non-trivial element of $\tau_{\pi}(G/N)$ of order $\pi^{i}$
for some $i\geq 1$ (such an element exists since $\tau_{\pi}(G/N)$ is
unbounded.) Set $k_{\ell}N=(k_{i}N)^{\pi^{i-\ell}}$ and note that $k_{\ell}N$
has order $\pi^{\ell}$ for $1\leq\ell\leq i-1.$ The unboundedness of
$\tau_{\pi}(G/N)$ allows us to find an element $k_{j}N$ of order $\pi^{j},$
where $j>i.$ Once again, if we put $k_{s}N=(k_{j}N)^{\pi^{j-s}},$ then
$k_{s}N$ has order $\pi^{s}$ for $i+1\leq s\leq j-1.$ We continue this process
and obtain a countable set of (distinct) elements $k_{1}N,\ k_{2}N,\ \ldots$
of $\tau_{\pi}(G/N)$ such that the order of each $k_{i}N$ is $\pi^{i}$.

Put $K/N=gp_{\mathbb{Q}[x]}(k_{1}N,\ k_{2}N,\ \ldots),$ and observe that $K/N$
is a $\pi$-torsion group with a countably infinite set of
$\mathbb{Q}[x]$-generators. There are two cases to consider, depending on
whether or not $K/N$ contains an element of infinite height.

Case (i): Suppose that $K/N$ has no non-trivial elements of infinite height.
By Theorem 2.14, $K/N$ is a direct product of countably many cyclic
$\mathbb{Q}[x]$-subgroups whose exponents need not be bounded. In this case,
there is a surjective $\mathbb{Q}[x]$-homomorphism $\varphi:K/N\rightarrow L$,
where $L$ is a $\pi$-quasicyclic $\mathbb{Q}[x]$-group. By the
$\mathbb{Q}[x]$-isomorphism theorems,

$L\cong_{\mathbb{Q}[x]}\frac{K/N}{P/N}\leq_{\mathbb{Q}[x]}\frac{G/N}{P/N}\cong_{\mathbb{Q}[x]}G/P,$

where $P/N$ is the kernel of $\varphi.$ Therefore, some
$\mathbb{Q}[x]$-quotient of $G$ contains a $\pi$-quasicyclic
$\mathbb{Q}[x]$-subgroup as asserted.

Case (ii): Assume that $aN$ is an element of $K/N$ of infinite height and
$aN\neq N.$ By Theorems 2.8 and 2.15, there is a $\mathbb{Q}[x]$-monomorphism
of $G/N$ into a divisible $\mathbb{Q}[x]$-group $D$ and $D$ is
$\mathbb{Q}[x]$-isomorphic to a direct product of copies of
$Q(\sigma^{\infty})$ for various primes $\sigma\in\mathbb{Q}[x]$ and copies of
the field of fractions of $\mathbb{Q}[x].$ This direct product must have a
$\pi$-quasicyclic factor since $aN$ has bounded order $\pi^{m}$ for some
$m\in\mathbb{N}.$ Thus, there exists a $\mathbb{Q}[x]$-homomorphism $\rho$ of
$D$ onto a $\pi$-quasicyclic factor (say $L$) whose kernel does not contain
$aN.$

We show that there is a $\mathbb{Q}[x]$-homomorphism of $G$ onto $L.$ Regard
$G/N$ as a $\mathbb{Q}[x]$-subgroup of $D$ and set $\psi=\rho|_{G/N}.$ We
claim that $\psi$ has unbounded image. To begin with, note that there exist
elements $x_{i}\in K$ ($i\in\mathbb{N}$) such that

$aN=(x_{1}N)^{\pi}=(x_{2}N)^{\pi^{2}}=\cdots=(x_{i}N)^{\pi^{i}}=\cdots=$

because $aN$ has infinite height. Since $aN\notin\ker\rho$,

$\rho(aN)=\psi(aN)=(\psi(x_{i}N))^{\pi^{i}}\neq 1$

for all $i\in\mathbb{N}$. Thus, the image of $\psi$ must be unbounded as
claimed. Now, since the only non-trivial proper $\mathbb{Q}[x]$-subgroups of
$L$ are bounded and $\psi(aN)\neq 1,$ the image of $\psi$ equals $L.$ And so,
if $\varphi:G\rightarrow G/N$ is the natural $\mathbb{Q}[x]$-homomorphism,
then $\psi\circ\varphi:G\rightarrow L$ is a surjective
$\mathbb{Q}[x]$-homomorphism. It follows from the $\mathbb{Q}[x]$-isomorphism
theorems that some $\mathbb{Q}[x]$-quotient of $G$ is
$\mathbb{Q}[x]$-isomorphic to a $\pi$-quasicyclic $\mathbb{Q}[x]$-subgroup.
This proves the assertion for Case (ii).

Next we prove by contradiction that if $G$ satisfies Condition ($B_{\omega}$),
then none of the $\mathbb{Q}[x]$-quotients of $G$ contain $\pi$-quasicyclic
$\mathbb{Q}[x]$-subgroups for all $\pi\in\omega.$ Suppose that this is not the
case, and there is a $\mathbb{Q}[x]$-quotient $G/N$ containing a
$\pi$-quasicyclic $\mathbb{Q}[x]$-subgroup, say $H/N$, for some
$\pi\in\omega$. Then $H/N\leq_{\mathbb{Q}[x]}\tau_{\pi}(G/N)$ since all
$\pi$-quasicyclic $\mathbb{Q}[x]$-groups are $\pi$-torsion. Since
$\tau_{\pi}(G/N)$ is bounded by hypothesis, so is $H/N.$ This contradicts the
fact that $H/N$ is an unbounded $\pi$-torsion $\mathbb{Q}[x]$-group. ∎

######  Theorem 3.4.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ An abelian
$\mathbb{Q}[x]$-group $G$ has property $\mathcal{G}_{\omega}$ if and only if
it satisfies Condition ($B_{\omega}$).

###### Proof.

Assume that $G$ does not satisfy Condition ($B_{\omega}$). By Proposition 3.3,
there exists $N\leq_{\mathbb{Q}[x]}G$ such that $G/N$ contains a
$\pi$-quasicyclic $\mathbb{Q}[x]$-subgroup for some $\pi\in\omega.$ Set
$\mathbb{P}=\mathbb{M}\setminus\\{\pi\\}.$ By Theorem 2.4, the family
$\left\\{\tau_{\beta}(G/N)\right\\}_{\beta\in\mathbb{P}}$ of
$\mathbb{Q}[x]$-subgroups of the $\mathbb{Q}[x]$-torsion subgroup of $G/N$ is
a direct product of its members. Put

$T/N=\prod_{\beta\in\mathbb{P}}\tau_{\beta}(G/N).$

If $\varphi:G/N\longrightarrow(G/N)/(T/N)$ is the natural
$\mathbb{Q}[x]$-homomorphism and $L/N$ is a $\pi$-quasicyclic
$\mathbb{Q}[x]$-subgroup of $G/N,$ then $\varphi(L/N)$ is also
$\pi$-quasicyclic because $L/N$ has trivial intersection with $T/N$ =
$\ker\varphi.$ Thus, by the $\mathbb{Q}[x]$-isomorphism theorems, $G/T$
contains a $\pi$-quasicyclic $\mathbb{Q}[x]$-subgroup. Hence, $G/T$ contains a
non-trivial $\mathbb{Q}[x]$-divisible $\mathbb{Q}[x]$-subgroup by Lemma 2.7.
By Theorem 2.11, $G/T$ is not residually finite type and, consequently, not
residually finite $\omega$-type. By Lemma 3.1, $T$ cannot be
$\mathcal{FT_{\omega}}$-separable in $G.$ On the other hand, $T$ is
$\omega^{\prime}$-isolated in $G$ because $\tau(G/T)$ is a $\pi$-torsion
group. To see why this is true, suppose that $g^{\alpha}\in T$ for some
$\alpha\in\mathbb{Q}[x]$ that is not an $\omega$-member. We claim that $g\in
T.$ Well, $g^{\alpha}\in T$ implies that $(gT)^{\alpha}=T$ in $G/T.$ Since
$\tau(G/T)$ is a $\pi$-torsion group, there exists $t\in\mathbb{N}$ such that
$(gT)^{\pi^{t}}=T.$ Since $\alpha$ is not an $\omega$-member and
$\pi\in\omega,$ there exist elements $a,\ b\in\mathbb{Q}[x]$ such that
$a\alpha+b\pi^{t}=1.$ Hence,

$gT=(gT)^{a\alpha+b\pi^{t}}=(gT)^{a\alpha}(gT)^{b\pi^{t}}=T.$

And so, $g\in T$ as claimed. Therefore, $G$ does not have property
$\mathcal{G}_{\omega}.$

Next suppose that $G$ satisfies Condition ($B_{\omega}$), and let $H$ be an
$\omega^{\prime}$-isolated $\mathbb{Q}[x]$-subgroup in $G.$ We claim that $H$
is $\mathcal{FT_{\omega}}$-separable or, equivalently (by Lemma 3.1), that
$G/H$ is residually $\mathcal{FT_{\omega}}.$ To simplify notation, we let
$\overline{K}$ denote the $\mathbb{Q}[x]$-quotient $K/H$ for
$H\leq_{\mathbb{Q}[x]}K$ and any $K\leq_{\mathbb{Q}[x]}G.$ Choose an element
$aH\in\overline{G}$ such that $aH\neq H.$ We divide the proof into 3 cases
which depend on the order of $aH.$

Case 1: Suppose that $aH$ has monic prime power order, say $\pi^{r}$ for some
monic prime $\pi\in\mathbb{Q}[x]$ and $r\in\mathbb{N}.$ Since $H$ is
$\omega^{\prime}$-isolated in $G$ and $aH\neq H$ it must be that
$\pi\in\omega.$ (If, instead, $\pi\in\omega^{\prime},$ then $a^{\pi^{r}}\in H$
would imply that $a\in H,$ a contradiction).

Let $\pi^{t}$ be the exponent of $\tau_{\pi}\left(\overline{G}\right)$ for
some $t\in\mathbb{N}$ (the exponent is bounded because $G$ satisfies Condition
($B_{\omega}$)). It is clear that
$\overline{G}^{\pi^{t}}\cap\tau_{\pi}\left(\overline{G}\right)=\\{H\\}.$
Hence, $aH\overline{G}^{\pi^{t}}\neq\overline{G}^{\pi^{t}}.$ Furthermore, the
$\pi$-torsion group $\overline{G}/\overline{G}^{\pi^{t}}$ is bounded and,
thus, $\mathbb{Q}[x]$-isomorphic to a direct product of $\pi$-torsion cyclic
$\mathbb{Q}[x]$-subgroups by Theorem 2.5. And so,
$\overline{G}/\overline{G}^{\pi^{t}}$ is residually $\mathcal{FT_{\pi}}$ by
Corollary 2.10. It follows that $\overline{G}$ is also residually
$\mathcal{FT_{\pi}}.$ Since $\pi\in\omega,$ we conclude that $\overline{G}$ is
residually $\mathcal{FT_{\omega}}.$

Case 2: Suppose that $aH$ has non-infinite order that is monic and not a power
of a prime, say $\pi^{r}\mu\in\mathbb{Q}[x],$ where $\pi\in\mathbb{Q}[x]$ is a
monic prime, $r\in\mathbb{N},$ and $\mu\in\mathbb{Q}[x]$ is monic and not
divisible by $\pi.$ As in Case 1, we have that $\pi\in\omega$; for otherwise,
$a^{\pi^{r}\mu}\in H$ would imply that $a^{\mu}\in H$ since $H$ is
$\omega^{\prime}$-isolated, and this would be impossible.

Set $\mathbb{P}=\mathbb{M}\setminus\\{\pi\\}.$ By Theorem 2.4 the
$\mathbb{Q}[x]$-subgroups $\tau_{\beta}\left(\overline{G}\right)$ form a
direct product for the various primes $\beta\in\mathbb{P}.$ Set

$N=\prod_{\beta\in\mathbb{P}}\tau_{\beta}\left(\overline{G}\right).$

We claim that $N$ is $\omega^{\prime}$-isolated in $\overline{G}$. To see this
assume that $(gH)^{\alpha}\in N$ for some $gH\in\overline{G}$ and
$\omega^{\prime}$-member $\alpha.$ Then $\pi$ is relatively prime to both
$\alpha$ and to the order of $(gH)^{\alpha}$. Hence, $\pi$ is relatively prime
to the order of $gH$ itself. This shows that $gH\in N$, and our claim holds.
Note also that $(aH)^{\pi^{r}}\in N$ since the order of $(aH)^{\pi^{r}}$ is
$\mu$, which is relatively prime to $\pi$. Hence $(aH)N$ has prime power order
in $\overline{G}/N$.

By Case 1, there exist $K\in\mathcal{FT_{\omega}}$ and an onto
$\mathbb{Q}[x]$-homomorphism

$\psi:\overline{G}/N\rightarrow K$

such that $\psi[(aH)N]\neq 1$. If $\Phi$ is the natural
$\mathbb{Q}[x]$-homomorphism from $\overline{G}$ onto $\overline{G}/N$, then
$\Phi\circ\psi$ is a $\mathbb{Q}[x]$-homomorphism from $\overline{G}$ onto $K$
that maps $aH$ to a non-trivial element in $K$. This completes Case 2.

Case 3: Let $aH$ have infinite order and set $C=gp_{\mathbb{Q}[x]}(aH).$ We
assert that $\overline{G}/C$ satisfies Condition ($B_{\omega}$). Well, if
$\overline{K}/C\leq_{\mathbb{Q}[x]}\overline{G}/C,$ then

$\frac{\overline{G}/C}{\overline{K}/C}\cong_{\mathbb{Q}[x]}\overline{G}/\overline{K}\cong_{\mathbb{Q}[x]}G/K.$

Since $G$ satisfies Condition ($B_{\omega}$), $\tau_{\pi}(G/K)$ is bounded for
any $\pi\in\omega.$ Hence
$\tau_{\pi}\left(\left(\overline{G}/C\right)/\left(\overline{K}/C\right)\right)$
is also bounded and, consequently, $\overline{G}/C$ satisfies Condition
($B_{\omega}$) as asserted. Thus, $\tau_{\pi}\left(\overline{G}/{C}\right)$ is
bounded for each $\pi\in\omega.$

Choose a specific prime $\alpha\in\omega.$ We prove that there exists a
natural number $r$ such that $aH\notin\overline{G}^{\alpha^{r}}.$ Suppose, on
the contrary, that $aH\in\overline{G}^{\alpha^{t}}$ for all $t\in\mathbb{N}.$
Then there exist $g_{1},\,g_{2},\,\ldots\in G$ such that

$aH=\left(g_{1}H\right)^{\alpha}=\left(g_{2}H\right)^{\alpha^{2}}=\cdots$

in $\overline{G}$. So, in $\overline{G}/C$, we have

$C=(aH)C=\left(g_{1}H\right)^{\alpha}C=\left(g_{2}H\right)^{\alpha^{2}}C=\cdots.$

Consequently,
$\left(g_{i}H\right)C\in\tau_{\alpha}\left(\overline{G}/C\right)$ for all
$i\in\mathbb{N}.$ We claim that the order of $(g_{i}H)C$ is precisely
$\alpha^{i}.$ This is obviously the case for $i=1$. Let $i>1$. Assume that the
order of $(g_{i}H)C$ is $\alpha^{k}$ for some $k<i$. Since
$(g_{i}H)^{\alpha^{k}}\in C=gp_{\mathbb{Q}[x]}(aH)$, there is a
$\mu\in\mathbb{Q}[x]$ such that $(g_{i}H)^{\alpha^{k}}=(aH)^{\mu}$. Then

$aH=(g_{i}H)^{\alpha^{i}}=\left((g_{i}H)^{\alpha^{k}}\right)^{\alpha^{i-k}}=(aH)^{\mu\alpha^{i-k}}.$

Hence, $(aH)^{\mu\alpha^{i-k}-1}=H$. Since $aH$ is of infinite order,
$\mu\alpha^{i-k}-1=0$, which is false. Thus, there exists $r\in\mathbb{N}$
such that $aH\notin\overline{G}^{\alpha^{r}}$, as claimed.

It is clear that $\overline{G}^{\alpha^{r}}$ is $\omega^{\prime}$-isolated in
$\overline{G}$, and that $(aH)\overline{G}^{\alpha^{r}}$ has order a power of
$\alpha$ in $\overline{G}/\overline{G}^{\alpha^{r}}$. Proceeding as in Case 2,
we can find a group $L$ in $\mathcal{FT_{\omega}}$ and a
$\mathbb{Q}[x]$-homomorphism from $\overline{G}$ onto $L$ that maps $aH$ non-
trivially. This completes the proof of the theorem. ∎

######  Definition 3.2.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ An abelian
$\mathbb{Q}[x]$-group $G$ is _$\omega$ -restricted_ if for every
$\mathbb{Q}[x]$-quotient $G/N$ of $G$ and every $\pi\in\omega$ the
$\pi$-primary component $\tau_{\pi}(G/N)$ of $G/N$ is finitely
$\mathbb{Q}[x]$-generated (hence, finite $\pi$-type).

The class of $\omega$-restricted abelian $\mathbb{Q}[x]$-groups will be
denoted by $\mathcal{A}_{\omega}.$ Note that every $\omega$-restricted abelian
$\mathbb{Q}[x]$-group satisfies Condition ($B_{\omega}$) since nilpotent
$\mathbb{Q}[x]$-powered groups of finite $\pi$-type are bounded.

######  Lemma 3.5.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ Every finitely
$\mathbb{Q}[x]$-generated abelian $\mathbb{Q}[x]$-group is
$\omega$-restricted.

###### Proof.

If $G$ is a finitely $\mathbb{Q}[x]$-generated abelian $\mathbb{Q}[x]$-group
and $N\leq_{\mathbb{Q}[x]}G,$ then $G/N$ is also finitely
$\mathbb{Q}[x]$-generated abelian. By Theorem 2.2, any
$\mathbb{Q}[x]$-subgroup of $G/N$ is finitely $\mathbb{Q}[x]$-generated. In
particular, $\tau_{\pi}(G/N)$ is finitely $\mathbb{Q}[x]$-generated. ∎

######  Definition 3.3.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ A nilpotent
$\mathbb{Q}[x]$-powered group is _$\omega$ -restricted_ if it has a finite
central $\mathbb{Q}[x]$-series with $\omega$-restricted abelian
$\mathbb{Q}[x]$-factors.

The class of $\omega$-restricted nilpotent $\mathbb{Q}[x]$-powered groups will
be denoted by $\mathcal{N}_{\omega}.$ Clearly,
$\mathcal{A}_{\omega}\subset\mathcal{N}_{\omega}.$

######  Lemma 3.6.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ Every finitely
$\mathbb{Q}[x]$-generated nilpotent $\mathbb{Q}[x]$-powered group is
$\omega$-restricted.

###### Proof.

Let $G$ be a finitely $\mathbb{Q}[x]$-generated nilpotent
$\mathbb{Q}[x]$-powered group. Suppose that

$\\{1\\}=G_{0}\leq_{\mathbb{Q}[x]}G_{1}\leq_{\mathbb{Q}[x]}\cdots\leq_{\mathbb{Q}[x]}G_{n}=G$

is any central $\mathbb{Q}[x]$-series. By Theorem 2.2, each $G_{i}$ is
finitely $\mathbb{Q}[x]$-generated. Thus, each $G_{i+1}/G_{i}$ is finitely
$\mathbb{Q}[x]$-generated abelian. The result follows from Lemma 3.5. ∎

We record a useful lemma. Its proof is omitted.

######  Lemma 3.7.

Let $G$ be a nilpotent $R$-powered group, and let $A,\ B,$ and $C$ be
$R$-subgroups of $G.$ If $B\unlhd_{R}A,$ then $(B\cap C)\unlhd_{R}(A\cap C)$
and

$\frac{A\cap C}{B\cap C}\cong_{R}\frac{B(A\cap C)}{B}.$

If $C\unlhd_{R}G$ as well, then $BC\unlhd_{R}AC$ and

$\frac{AC}{BC}\cong_{R}\frac{A}{B(A\cap C)}.$

######  Proposition 3.8.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ Every
$\mathbb{Q}[x]$-subgroup and $\mathbb{Q}[x]$-quotient of an
$\mathcal{A}_{\omega}$-group ($\mathcal{N}_{\omega}$-group) is an
$\mathcal{A}_{\omega}$-group ($\mathcal{N}_{\omega}$-group).

###### Proof.

(i) Let $G\in\mathcal{A}_{\omega}$ and $H\leq_{\mathbb{Q}[x]}G.$ Suppose
$K\leq_{\mathbb{Q}[x]}H$ and choose $\pi\in\omega.$ Clearly,
$\tau_{\pi}(H/K)\leq_{\mathbb{Q}[x]}\tau_{\pi}(G/K).$ Since $\tau_{\pi}(G/K)$
is finitely $\mathbb{Q}[x]$-generated, so is $\tau_{\pi}(H/K)$ by Theorem 2.2.
Hence, $H\in\mathcal{A}_{\omega}.$

Next let $N\leq_{\mathbb{Q}[x]}G$ and $M/N\leq_{\mathbb{Q}[x]}G/N.$ For any
$\pi\in\omega,$ $\tau_{\pi}(G/M)$ is finitely $\mathbb{Q}[x]$-generated
because $G\in\mathcal{A}_{\omega}.$ Since
$G/M\cong_{\mathbb{Q}[x]}(G/N)/(M/N),$ we have $G/N\in\mathcal{A}_{\omega}.$

(ii) Now suppose that $G\in\mathcal{N}_{\omega}$ with a central
$\mathbb{Q}[x]$-series

$\\{1\\}=G_{0}\leq_{\mathbb{Q}[x]}G_{1}\leq_{\mathbb{Q}[x]}\cdots\leq_{\mathbb{Q}[x]}G_{n}=G$

such that each $G_{i+1}/G_{i}$ is $\omega$-restricted abelian. If
$H\leq_{\mathbb{Q}[x]}G,$ then

$\\{1\\}=G_{0}\cap H\leq_{\mathbb{Q}[x]}G_{1}\cap
H\leq_{\mathbb{Q}[x]}\cdots\leq_{\mathbb{Q}[x]}G_{n}\cap H=H$

is a central $\mathbb{Q}[x]$-series for $H.$ By Lemma 3.7, $(G_{i+1}\cap
H)/(G_{i}\cap H)$ is $\mathbb{Q}[x]$-isomorphic to a $\mathbb{Q}[x]$-subgroup
of $G_{i+1}/G_{i}.$ Since $G_{i+1}/G_{i}\in\mathcal{A}_{\omega},$ so is
$(G_{i+1}\cap H)/(G_{i}\cap H)$ by (i) above. Hence,
$H\in\mathcal{N}_{\omega}.$

Next let $N\unlhd_{\mathbb{Q}[x]}G.$ The quotient $\mathbb{Q}[x]$-group $G/N$
has a central $\mathbb{Q}[x]$-series

$\\{1\\}=G_{0}N/N\leq_{\mathbb{Q}[x]}G_{1}N/N\leq_{\mathbb{Q}[x]}\cdots\leq_{\mathbb{Q}[x]}G_{n}N/N=G/N.$

By Lemma 3.7 and a $\mathbb{Q}[x]$-isomorphism theorem,
$(G_{i+1}N/N)/(G_{i}N/N)$ is $\mathbb{Q}[x]$-isomorphic to
$(G_{i+1}/G_{i})/(G_{i}(G_{i+1}\cap N)/G_{i}).$ Hence,
$(G_{i+1}N/N)/(G_{i}N/N)$ is $\mathbb{Q}[x]$-isomorphic to a
$\mathbb{Q}[x]$-quotient of $G_{i+1}/G_{i}$ and, consequently,
$G/N\in\mathcal{N}_{\omega}$ by (i). ∎

Notation: Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ The class of
nilpotent $\mathbb{Q}[x]$-powered groups having a finite subnormal
$\mathbb{Q}[x]$-series with $\omega$-restricted abelian
$\mathbb{Q}[x]$-factors will be denoted by $\mathcal{S}_{\omega}.$

Note that for a set of primes $\omega$ of $\mathbb{Q}[x],$ we have that
$\mathcal{N}_{\omega}\subset\mathcal{S}_{\omega}$ since every central
$\mathbb{Q}[x]$-series is a subnormal $\mathbb{Q}[x]$-series.

######  Proposition 3.9.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ Every
$\mathbb{Q}[x]$-subgroup and $\mathbb{Q}[x]$-quotient of an
$\mathcal{S}_{\omega}$-group is an $\mathcal{S}_{\omega}$-group.

###### Proof.

Replace “central” with “subnormal” in the proof of Proposition 3.8. ∎

######  Proposition 3.10.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ An abelian
$\mathbb{Q}[x]$-group is $\omega$-restricted if and only if it is in
$\mathcal{S}_{\omega}.$

###### Proof.

Let $G$ be an abelian $\mathbb{Q}[x]$-group in $\mathcal{S}_{\omega},$ and let
$H\leq_{\mathbb{Q}[x]}G.$ By Proposition 3.9, $G/H\in\mathcal{S}_{\omega}$
and, thus, $\tau_{\beta}(G/H)\in\mathcal{S}_{\omega}$ for any
$\beta\in\omega.$ Fix a prime $\pi\in\omega,$ and put $T=\tau_{\pi}(G/H).$ We
claim that $T$ is finitely $\mathbb{Q}[x]$-generated, thus proving that $G$ is
an $\mathcal{A}_{\omega}$-group.

Since $T\in\mathcal{S}_{\omega},$ it has a $\mathbb{Q}[x]$-series

(1)
$\\{1\\}=T_{0}\leq_{\mathbb{Q}[x]}T_{1}\leq_{\mathbb{Q}[x]}\cdots\leq_{\mathbb{Q}[x]}T_{k}=T$

with $\omega$-restricted abelian $\mathbb{Q}[x]$-factors. Thus, each
$\tau_{\pi}(T_{i+1}/T_{i})$ is finitely $\mathbb{Q}[x]$-generated. Since each
term in (1) is $\pi$-torsion, each $T_{i+1}/T_{i}$ is $\pi$-torsion as well.
Hence, $\tau_{\pi}(T_{i+1}/T_{i})=T_{i+1}/T_{i}$ and thus, $T_{i+1}/T_{i}$ is
finitely $\mathbb{Q}[x]$-generated. And so, $T$ is finitely
$\mathbb{Q}[x]$-generated as claimed. The converse is clear. ∎

######  Corollary 3.11.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ Every abelian
$\mathbb{Q}[x]$-group in $\mathcal{N}_{\omega}$ is $\omega$-restricted.

###### Proof.

This follows from Proposition 3.10 and the fact that
$\mathcal{N}_{\omega}\subset\mathcal{S}_{\omega}.$ ∎

Our next goal is to prove that for any set of primes $\omega$ in
$\mathbb{Q}[x],$ every $\omega$-restricted nilpotent $\mathbb{Q}[x]$-powered
group has property $\mathcal{G}_{\omega}$ for its family of normal
$\mathbb{Q}[x]$-subgroups. First we prove some preliminary results.

######  Proposition 3.12.

Let $\omega$ be a set of primes in $\mathbb{Q}[x]$ and
$G\in\mathcal{N}_{\omega}.$ If $\mu$ is any $\omega$-member, then all
$\mathbb{Q}[x]$-factors of any central $\mathbb{Q}[x]$-series of $G/G^{\mu}$
are finite $\omega$-type. Consequently, $G/G^{\mu}$ is finite $\omega$-type.

###### Proof.

As usual, we may assume that all primes in $\omega$ and all $\omega$-members
are monic. Let $\mu$ be an $\omega$-member and denote the set of monic prime
divisors of $\mu$ by $\sigma.$ Clearly, $\sigma$ is a finite set and $\mu$ is
a $\sigma$-member. Define the set $\sigma^{\prime}$ by
$\sigma^{\prime}=\omega\setminus\sigma.$ Suppose that

$\\{1\\}=G^{\mu}/G^{\mu}\leq_{\mathbb{Q}[x]}G_{1}/G^{\mu}\leq_{\mathbb{Q}[x]}\cdots\leq_{\mathbb{Q}[x]}G_{n}/G^{\mu}=G/G^{\mu}$

is an arbitrary central $\mathbb{Q}[x]$-series for $G/G^{\mu},$ and let
$A=(G_{i+1}/G^{\mu})/(G_{i}/G^{\mu})$ be any one of its
$\mathbb{Q}[x]$-factors. We claim that $A$ is finite $\omega$-type.

By Proposition 3.8, $G/G^{\mu}\in\mathcal{N}_{\omega}$ because
$G\in\mathcal{N}_{\omega}.$ It follows from Proposition 3.8 and Corollary 3.11
that $A\in\mathcal{A}_{\omega}.$ Hence, $\tau_{\pi}(A)$ is finitely
$\mathbb{Q}[x]$-generated for every $\pi\in\omega.$ Now, $A$ is a
$\sigma$-torsion group because $G_{k}/G^{\mu}$ is $\sigma$-torsion for every
$k=1,\ 2,\ \ldots,\ n.$

Thus, by Theorem 2.4,

$\tau_{\omega}(A)=\prod_{\pi\in\sigma}\tau_{\pi}(A)\times\prod_{\pi^{\prime}\in\sigma^{\prime}}\tau_{\pi^{\prime}}(A)\cong_{\mathbb{Q}[x]}\prod_{\pi\in\sigma}\tau_{\pi}(A).$

Since $\sigma$ is finite, $\tau_{\omega}(A)$ is finitely
$\mathbb{Q}[x]$-generated. Therefore, $A$ is finite $\omega$-type as claimed.
Since there are finitely many terms in the central $\mathbb{Q}[x]$-series,
$G/G^{\mu}$ is finite $\omega$-type. ∎

######  Proposition 3.13.

Suppose that $\omega$ is a set of primes in $\mathbb{Q}[x]$ and
$G\in\mathcal{N}_{\omega}.$ If $1\neq g\in G$ has bounded order an
$\omega$-member, then $g\notin G^{\beta}$ for some $\omega$-member $\beta.$

###### Proof.

Suppose that $g$ has bounded order an $\omega$-member $\mu,$ and let $\sigma$
be the set of monic prime divisors of $\mu.$ Note that $\sigma$ is a finite
set and $\mu$ is a $\sigma$-member. Since $G\in\mathcal{N}_{\omega}$ and
$\sigma\subseteq\omega,$ there exists a central $\mathbb{Q}[x]$-series

$\\{1\\}=G_{0}\leq_{\mathbb{Q}[x]}G_{1}\leq_{\mathbb{Q}[x]}\cdots\leq_{\mathbb{Q}[x]}G_{n}=G$

of $G$ such that each $\mathbb{Q}[x]$-factor $\tau_{\pi}(G_{i+1}/G_{i})$ is
finitely $\mathbb{Q}[x]$-generated for every $\pi\in\sigma.$ By Theorem 2.4
and the fact that $\sigma$ is a finite set, we see that

$\prod_{\pi\in\sigma}\tau_{\pi}(G_{i+1}/G_{i})=\tau_{\sigma}(G_{i+1}/G_{i})$

is finitely $\mathbb{Q}[x]$-generated. Abbreviate the exponent of a nilpotent
$\mathbb{Q}[x]$-powered group by “exp” and define

$\beta=\prod_{i=0}^{n-1}\hbox{exp}\left(\tau_{\sigma}(G_{i+1}/G_{i}))\right).$

We claim that $g\notin G^{\beta}.$ According to Theorem 2.6, every element of
$G^{\beta}$ can be written as a power of $\beta.$ If it were true that $g\in
G^{\beta},$ then there would exist $h\in G$ such that $h^{\beta}=g.$ Hence,
$h^{\beta\mu}=g^{\mu}=1.$ Now, $\beta\mu$ is a $\sigma$-member since $\beta$
and $\mu$ are both $\sigma$-members. Thus, $h^{\beta\mu}=1$ implies that
$h\in\tau_{\sigma}(G).$ Consequently, $h^{\beta}=1.$ Therefore, $g=1,$ a
contradiction. ∎

######  Theorem 3.14.

Let $\omega$ be a set of primes in $\mathbb{Q}[x]$ and
$G\in\mathcal{N}_{\omega}.$ Then, every $\mathbb{Q}[x]$-torsion element of $G$
is $\omega$-torsion if and only if $G$ is residually $\mathcal{FT_{\omega}}.$

###### Proof.

Suppose $G$ is residually $\mathcal{FT_{\omega}},$ and let $g\neq 1$ be a
$\mathbb{Q}[x]$-torsion element. There exists $H\in\mathcal{FT_{\omega}}$ and
a $\mathbb{Q}[x]$-homomorphism $\varphi$ of $G$ onto $H$ such that
$\varphi(g)\neq 1.$ Since $g\in\tau(G),$ there exists
$0\neq\alpha\in\mathbb{Q}[x]$ such that $g^{\alpha}=1.$ Hence,
$1=\varphi(g^{\alpha})=[\varphi(g)]^{\alpha}.$ Since $\varphi(g)\neq 1$ and
$H$ is $\omega$-torsion, $\alpha$ must be an $\omega$-member. Therefore,
$\tau(G)$ is $\omega$-torsion.

Conversely, assume that $\tau(G)$ is an $\omega$-torsion group. We prove that
$G$ is residually $\mathcal{FT_{\omega}}$ by induction on the class $c$ of
$G.$

  * •

If $G$ has class $1,$ then it is abelian. By assumption, $G$ is
$\omega^{\prime}$-torsion free. Since
$\tau(G)\cong_{\mathbb{Q}[x]}\tau(G)/\\{1\\},$ the trivial
$\mathbb{Q}[x]$-subgroup of $G$ is $\omega^{\prime}$-isolated in $G$ by Lemma
2.13. Now, $G$ is $\omega$-restricted by Corollary 3.11. Therefore, $G$ also
satisfies Condition ($B_{\omega}),$ and thus, has property
$\mathcal{G}_{\omega}$ by Theorem 3.4. Hence, $\\{1\\}$ is
$\mathcal{FT_{\omega}}$-separable in $G.$ By Lemma 3.1, $G$ is residually
$\mathcal{FT_{\omega}}.$

  * •

Assume that $G$ has class $c>1$ and that the result is true for all
$\omega$-restricted nilpotent $R$-powered groups of smaller class. Let $1\neq
g\in G.$ We claim that there exists a nilpotent $\mathbb{Q}[x]$-powered group
of finite $\omega$-type and a $\mathbb{Q}[x]$-homomorphism $\varphi$ of $G$
onto this group such that $\varphi(g)\neq 1.$

    1. (1)

If $g\notin Z(G),$ then $gZ(G)\neq Z(G)$ in $G/Z(G).$ Now, $G/Z(G)$ is
nilpotent of class less than $c.$ Furthermore, $\tau(G/Z(G))$ is an
$\omega$-torsion group because $\tau(G)$ is. By induction, there exists
$L\in\mathcal{FT_{\omega}}$ and a $\mathbb{Q}[x]$-homomorphism $\varphi$ from
$G/Z(G)$ onto $L$ such that $\varphi(gZ(G))\neq 1.$ Hence, if
$\psi:G\rightarrow G/Z(G)$ is the natural $\mathbb{Q}[x]$-homomorphism, then
$\varphi\circ\psi$ satisfies the claim.

    2. (2)

Suppose $1\neq g\in Z(G).$ Clearly, $\tau(Z(G))$ is $\omega$-torsion since
$\tau(G)$ is $\omega$-torsion. By Proposition 3.8,
$Z(G)\in\mathcal{N}_{\omega}.$ Hence, $Z(G)\in\mathcal{A}_{\omega}$ by
Corollary 3.11. By the induction hypothesis, there is a
$\mathbb{Q}[x]$-subgroup $N$ of $Z(G)$ such that
$Z(G)/N\in\mathcal{FT_{\omega}}$ and $g\notin N.$ Consequently, $gN\neq N$ and
$(gN)^{\alpha}=N$ in $G/N$ for some $\omega$-member $\alpha.$ Since
$G/N\in\mathcal{N}_{\omega},$ there exists an $\omega$-member $\beta$ such
that $gN\notin(G/N)^{\beta}$ by Proposition 3.13. And so,
$(G/N)/(G/N)^{\beta}\in\mathcal{FT_{\omega}}$ by Proposition 3.12. If

$\varphi_{1}:G\rightarrow G/N\hbox{ \ and \
}\varphi_{2}:G/N\rightarrow\frac{G/N}{(G/N)^{\beta}}$

are the natural $\mathbb{Q}[x]$-homomorphisms, then
$\varphi=\varphi_{2}\circ\varphi_{1}$ satisfies the claim.

∎

We are now prepared to prove the following:

######  Theorem 3.15.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ Every $\omega$-restricted
nilpotent $\mathbb{Q}[x]$-powered group has property $\mathcal{G}_{\omega}$
for its family of normal $\mathbb{Q}[x]$-subgroups.

###### Proof.

Let $G\in\mathcal{N}_{\omega}$ and assume that $H$ is an
$\omega^{\prime}$-isolated normal $\mathbb{Q}[x]$-subgroup of $G.$ By Lemma
2.13, $G/H$ is $\omega^{\prime}$-torsion-free. Since
$G\in\mathcal{N}_{\omega},$ we have $G/H\in\mathcal{N}_{\omega}$ by
Proposition 3.8. By Theorem 3.14, $G/H$ is residually $\mathcal{FT_{\omega}}$
because $\tau(G/H)$ is $\omega$-torsion. And so, $H$ is
$\mathcal{FT_{\omega}}$-separable in $G$ by Lemma 3.1. ∎

######  Theorem 3.16.

Let $\omega$ be a set of primes in $\mathbb{Q}[x].$ If $G$ is a
$\mathbb{Q}[x]$-torsion-free nilpotent $\mathbb{Q}[x]$-powered group and has
property $\mathcal{G}_{\omega},$ then each $\mathbb{Q}[x]$-factor group
$\zeta_{i+1}G/\zeta_{i}G$ of its upper central $\mathbb{Q}[x]$-series has
property $\mathcal{G}_{\omega}.$

By Theorem 3.4, $\zeta_{i+1}G/\zeta_{i}G$ also satisfies Condition
($B_{\omega}$).

###### Proof.

We begin by observing that $\zeta_{i}G$ is $\mathbb{Q}[x]$-isolated in $G$
because $G$ is $\mathbb{Q}[x]$-torsion-free (see [10]). If
$H\unrhd_{\mathbb{Q}[x]}\zeta_{i}G$ for some $H\leq_{\mathbb{Q}[x]}G,$ then it
easily verified that $H$ is $\omega^{\prime}$-isolated in $G$ whenever
$H/\zeta_{i}G$ is an $\omega^{\prime}$-isolated $\mathbb{Q}[x]$-subgroup in
$\zeta_{i+1}G/\zeta_{i}G.$

We claim that $H/\zeta_{i}G$ is $\mathcal{FT_{\omega}}$-separable in
$\zeta_{i+1}G/\zeta_{i}G.$ By hypothesis, we know that $H$ is
$\mathcal{FT_{\omega}}$-separable in $G.$ Thus, for each
$g\in\zeta_{i+1}G\setminus H,$ there exists a normal $\mathbb{Q}[x]$-subgroup
$N$ of $G$ (which depends on $g$) such that $G/N\in\mathcal{FT_{\omega}}$ and
$g\notin HN.$ It is clear that

$\frac{N\zeta_{i}G\cap\zeta_{i+1}G}{\zeta_{i}G}\unlhd_{\mathbb{Q}[x]}\frac{\zeta_{i+1}G}{\zeta_{i}G}\hbox{
\ \ and \ \
}g\zeta_{i}G\notin\frac{H}{\zeta_{i}G}\frac{(N\zeta_{i}G\cap\zeta_{i+1}G)}{\zeta_{i}G}.$

Using the $\mathbb{Q}[x]$-isomorphism theorems, we have

$\displaystyle\frac{\zeta_{i+1}G/\zeta_{i}G}{(N\zeta_{i}G\cap\zeta_{i+1}G)/\zeta_{i}G}$
$\displaystyle\cong_{\mathbb{Q}[x]}$
$\displaystyle\frac{\zeta_{i+1}G}{N\zeta_{i}G\cap\zeta_{i+1}G}\cong_{\mathbb{Q}[x]}\frac{\zeta_{i+1}G(N\zeta_{i}G)}{N\zeta_{i}G}$
$\displaystyle\cong_{\mathbb{Q}[x]}$
$\displaystyle\frac{N\zeta_{i+1}G/N}{N\zeta_{i}G/N}\leq_{\mathbb{Q}[x]}\frac{G/N}{N\zeta_{i}G/N}.$

Thus, $(\zeta_{i+1}G/\zeta_{i}G)/((N\zeta_{i}G\cap\zeta_{i+1}G)/\zeta_{i}G)$
is $\mathbb{Q}[x]$-isomorphic to a $\mathbb{Q}[x]$-subgroup of a
$\mathbb{Q}[x]$-quotient of $G/N$ and, thus, finite $\omega$-type by Corollary
2.3. Since $g$ was chosen arbitrarily, $H/\zeta_{i}G$ is
$\mathcal{FT_{\omega}}$-separable in $\zeta_{i+1}G/\zeta_{i}G.$ This proves
the claim and completes the proof. ∎

Open Problem: Find other classes of nilpotent $\mathbb{Q}[x]$-powered groups
for which every $\omega$-restricted nilpotent $\mathbb{Q}[x]$-powered group
has property $\mathcal{G}_{\omega}$ for its family of
$\mathbb{Q}[x]$-subgroups contained in each of these classes.

## References

  * [1] V. G. Bardakov, On D. I. Modavanskii’s question about $p$-separable subgroups of a free group, Siberian Math. J., 45, No. 3, 416 - 419 (2004)

  * [2] N. Bourbaki, Algebra II. Chapters 4-7, Elements of mathematics, Springer (1990)

  * [3] A. E. Clement, S. Majewicz, and M. Zyman, The Theory of Nilpotent Groups, Birkhauser (2017)

  * [4] P. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, MR0283083 (1969)

  * [5] M. I. Kargapolov, V. N. Remeslennikov, N. S. Romanovskii, V. A. Roman’kov, and V. A. Churkin, Algorithmic problems for $\sigma$-power groups, J. Algebra and Logic (Translated), Vol. 8 (6), 364 - 373 (1969)

  * [6] E. D. Loginova, Residual finiteness of the free product of two groups with commuting subgroups, Siberian Math. J., 40, No. 2, 341 - 350 (1999)

  * [7] S. Majewicz, Nilpotent $\mathbb{Q}[x]$-powered groups, (Amer. Math. Soc. Contemporary Math.), 421 (2006)

  * [8] S. Majewicz and M. Zyman, Power-commutative nilpotent $R$-powered groups, Groups-Complexity-Cryptology, Vol. 1, No. 2, 297 - 309 (2009)

  * [9] S. Majewicz and M. Zyman, On the extraction of roots in exponential $A$-groups, Groups Geom. Dyn. 4, 835 - 846 (2010)

  * [10] S. Majewicz and M. Zyman, On the extraction of roots in exponential $A$-groups II, Comm. in Alg., 40, 64 - 86 (2012)

  * [11] S. Majewicz and M. Zyman, Localization of nilpotent $R$-powered groups, J. Group Theory, 15, 119 - 135 (2012)

  * [12] S. Majewicz and M. Zyman, Separability properties of nilpotent $\mathbb{Q}[x]$-powered groups, Elementary Theory of Groups and Group Rings, and Related Topics, 203 - 218 (2020)

  * [13] E. V. Sokolov, Separability of subgroups of nilpotent groups in the class of finite $\pi$-groups, Siberian Math. J., 55, No. 6, 1126 - 1132 (2004)

  * [14] R. B. Warfield, Jr., Nilpotent Groups, Lecture Notes in Mathematics (513), Springer-Verlag (1976)


# Quantum simulation of perfect state transfer on weighted cubelike graphs

Jaideep Mulherkar
<EMAIL_ADDRESS>Rishikant Rajdeepak
<EMAIL_ADDRESS>V. Sunita
<EMAIL_ADDRESS>
###### Abstract

A continuous-time quantum walk on a graph evolves according to the unitary
operator $e^{-iAt}$, where $A$ is the adjacency matrix of the graph. Perfect
state transfer (PST) in a quantum walk is the transfer of a quantum state from
one node of a graph to another node with $100\%$ fidelity. It can be shown
that the adjacency matrix of a cubelike graph is a finite sum of tensor
products of Pauli $X$ operators. We use this fact to construct an efficient
quantum circuit for the quantum walk on cubelike graphs. In [5, 15], a
characterization of integer weighted cubelike graphs is given that exhibit
periodicity or PST at time $t=\pi/2$. We use our circuits to demonstrate PST
or periodicity in these graphs on IBM’s quantum computing platform [1, 10].

keywords: Continuous-time quantum walk, Perfect state transfer, Periodicity,
Quantum circuits.

##  1 Introduction

A quantum random walk is the quantum analogue of a classical random walk [12,
18, 19]. The study of classical random walks has led to many applications in
science and engineering, such as in the study of randomized algorithms and a
sampling approach called Markov chain Monte Carlo in computer science, in the
study of social networks, in the behavior of stock prices in finance, in
models of diffusion and study of polymers in Physics, and the motion of motile
bacteria in biology. In [3, 7], the first models for quantum random walks were
proposed. Since then, quantum walks have been a source of intense study.
Researchers observed that there are some startling differences between
classical and quantum walks. For example, a quantum walk on a one-dimensional
lattice spreads quadratically faster than a classical walk [16]. Quantum walks
on cubelike graphs, such as the hypercubes, hit exponentially faster to the
antipodal vertex as compared to classical counterparts [13].

Quantum walks on graphs are of two types: discrete and continuous. In the
discrete case, a graph is associated with a Hilbert space of dimension
$N\times\Delta$, where $N$ is the number of vertices, and $\Delta$ is the
maximum degree of the graph. In the continuous case, a graph is associated
with a Hilbert space of dimension $N$, and the evolution of the system is
described by $e^{\iota tA}$, where $A$ is the adjacency matrix of the graph
and $t$ is real. An essential feature of a quantum walk is a quantum state
transfer from one vertex to another with high fidelity. When the transfer
occurs with $100\%$ fidelity, it is called perfect state transfer (PST). Some
of the excellent survey papers on graph families that admit PST are [9, 8].
Among these graphs, cubelike graphs are the most famous ones that have been
researched thoroughly for determining the existence and finding the pair of
vertices admitting perfect state transfer in constant time [4, 6]. Notice that
all cubelike graphs do not allow perfect state transfer. The study of PST on
weighted graphs has been less studied. Recently, weighted abelian Cayley
graphs have been characterized that exhibit PST [5].

In this paper, we look at the implementation of Perfect state transfer on
weighed cubelike graphs. Some of the efficient implementation of quantum walks
are described in [2, 11, 14, 20, 21]. It can be shown that the adjacency
matrix of a cubelike graph is the sum of the tensor products of Pauli $X$
operators. One then observes that generating efficient quantum circuits for
quantum walks can then be done by quantum hamiltonian simulation techniques
that have been described in [17]. We use quantum simulation techniques to
verify the theoretical results of PST on weighted cubelike graphs.

##  2 Preliminaries

An undirected weighted graph $\Gamma$ consists of a triplet $(V,E,f)$, where
$V$ is a non-empty set whose elements are called vertices, $E$ is a set of
edges, where an edge is an unordered tuple of vertices, and $f:V\times
V\rightarrow\mathbb{R}$ is a weight function that assigns non-zero real
weights to edges. If $\Gamma$ is finite, then its adjacency matrix $A$ is
defined by;

$A_{u,v}=f((u,v)),\qquad(u,v)\in V\times V.$

The adjacency matrix $A$ is real and symmetric. A tuple $(u,u)$ is a loop if
its weight is non-zero. If $f((u,u))=0$ for all $u\in V$, then the diagonal
entries of $A$ are zero and the graph has no loops. A graph family of interest
is a weighted cubelike graph which is defined as follows.

######  Definition 2.1.

Let $f$ be a real-valued function over a finite Boolean group
$\mathbb{Z}_{2}^{n}$ of dimension $n>0$. A cubelike graph, denoted by
$Cay(\mathbb{Z}_{2}^{n},f)$, is a graph with vertex-set $\mathbb{Z}_{2}^{n}$,
and the weight of a pair $(u,v)$ of vertices is given by $f(u\oplus v)$, where
$\oplus$ denotes the group addition, i.e., componentwise addition modulo 2.
The adjacency matrix $A$ of $Cay(\mathbb{Z}_{2}^{n},f)$ is given by;

$A_{u,v}=f(u\oplus v),\;u,v\in V.$

An equivalent definition for an unweighted cubelike graph is given by; let
$\Omega_{f}=\\{u\in\mathbb{Z}_{2}^{n}:f(u)=1\\}$, then two vertices $u$ and
$v$ are adjacent if $u\oplus v\in\Omega_{f}$. The cubelike graph, in this
case, is denoted by $Cay(\mathbb{Z}_{2}^{n},\Omega_{f})$, see Fig. 2 and Fig.
2.

$000$$001$$011$$010$$100$$101$$111$$110$ Figure 1:
$Cay(\mathbb{Z}_{2}^{3},\\{001,010,100\\})$

$000$$001$$011$$010$$100$$101$$111$$110$ Figure 2:
$Cay(\mathbb{Z}_{2}^{3},\\{001,010,011,100,111\\})$

###  2.1 Continuous-time quantum walk

Let $\Gamma$ be an undirected and weighted graph with or without loops and $A$
be the adjacency matrix. A quantum walk on $\Gamma$ is described by an
evolution of the quantum system associated with the graph. Suppose the graph
has $N$ vertices, then it is associated with a Hilbert space
$\mathcal{H}_{P}\cong\mathbb{C}^{N}$ of dimension $N$, called the position
space, and the computational basis is represented by;

$\\{\ket{v}:v\mbox{ is a vertex in }\Gamma\\}.$

The continuous-time quantum walk (CTQW) on $\Gamma$ is described by the
transition matrix $\mathcal{U}(t)=e^{-\iota tA}$, where $\iota=\sqrt{-1}$,
that operates on the position space $\mathcal{H}_{P}$. In other words, if
$\ket{\psi(0)}$ is the initial state of the quantum system associated with the
graph, then the state of the system after time $t$ is given by

$\ket{\psi(t)}=e^{-\iota tA}\ket{\psi(0)}.$

######  Definition 2.2.

A graph is said to admit perfect state transfer (PST) if the quantum walker
beginning at some vertex $u$ reaches a distinct vertex $v$ with probability
$1$, i.e., for some positive real $\tau$ and scalar $\lambda$

$|\braket{v}{e^{-\iota\tau A}}{u}|=|\lambda|=1.$

Alternatively, we say perfect state transfer occurs from the vertex $u$ to the
vertex $v$. If $u=v$, we say the graph is periodic at $u$ with period $\tau$,
and if the graph is periodic at every vertex with the same period $\tau$ then,
the graph is periodic.

######  Example 2.3.

Consider the graph on the cycle of size 4, see Fig. 3, with the adjacency
matrix $A$ given by;

$A=\begin{bmatrix}0&1&1&0\\\ 1&0&0&1\\\ 1&0&0&1\\\ 0&1&1&0\end{bmatrix}.$

Then, the transition matrix at time $t=\pi/2$ is

$\mathcal{U}(t=\frac{\pi}{2})=\begin{bmatrix}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\end{bmatrix}.$

Thus, perfect state transfer occurs between the pairs $\\{1,4\\}$ and
$\\{2,3\\}$, both in time $\frac{\pi}{2}$. The graph is periodic with period
$\pi$.

1243 Figure 3: PST occurs between the pairs {1,4} and {2,3} with time
$\frac{\pi}{2}$, and the graph is periodic with period $\pi$.

###  2.2 Decomposition of the adjacency matrix of weighted cubelike graph

####  2.2.1 Group representations

An $m$-degree representation of a finite group $G$ is a homomorphism $\rho$
from $G$ into the general linear group $GL(V)$ of an $m$-dimensional vector
space $V$ over the field $\mathbb{F}$, where $\mathbb{F}$ is a complex or real
field. Since $GL(V)$ is isomorphic to $GL_{m}(\mathbb{F})$, the general linear
group of degree $m$ that consists of $m\times m$ invertible matrices, an
equivalent definition for the group representation is the group homomorphism

$\rho:G\rightarrow GL_{m}(\mathbb{F}).$

The group algebra $\mathbb{C}[G]$ is an inner product space whose vectors are
formal linear combinations of the group elements, i.e.,

$\mathbb{C}[G]=\left\\{\sum_{g\in
G}\lambda_{g}g:\lambda_{g}\in\mathbb{C}\right\\},$

with the vector addition, the scalar multiplication, and the inner product
defined by;

$\begin{split}\sum_{g\in G}\lambda_{g}g+\sum_{g\in G}\mu_{g}g&=\sum_{g\in
G}(\lambda_{g}+\mu_{g})g,\qquad\mbox{(addition)},\\\ \lambda\sum_{g\in
G}\lambda_{g}g&=\sum_{g\in G}(\lambda\lambda_{g})g\qquad\mbox{(scalar
multiplication)},\\\ \left\langle\sum_{g\in G}\lambda_{g}g,\sum_{g\in
G}\mu_{g}g\right\rangle&=\sum_{g\in
G}\lambda_{g}\bar{\mu}_{g},\qquad\mbox{(inner product)}.\end{split}$

The regular representation on $G$, $\rho_{reg}:G\rightarrow
GL(\mathbb{C}[G])$, is defined by;

$\rho_{reg}(x)\left(\sum_{g\in G}\lambda_{g}g\right)=\sum_{g\in
G}\lambda_{g}(xg)=\sum_{y\in G}\lambda_{x^{-1}y}y.$

####  2.2.2 The decomposition

If $G=\mathbb{Z}_{2}^{n}$, then for $x\in\mathbb{Z}_{2}^{n}$ the regular
representation acts on $\mathbb{Z}_{2}^{n}$ as

$\rho_{reg}(x)y=x\oplus y=(x_{1}\oplus y_{1},\dots,x_{n}\oplus y_{n}),\qquad
y\in\mathbb{Z}_{2}^{n}.$

Let $X$, $Y$ and $Z$ denote the three Pauli matrices that acts on the
computational basis $\\{\ket{0},\ket{1}\\}$ of the two dimensional Hilbert
space $\mathbb{C}^{2}$ as

$X\ket{a}=\ket{a\oplus 1},\;Y\ket{a}=(-1)^{a}\iota\ket{a\oplus
1},\;Z\ket{a}=(-1)^{a}\ket{a},\;a\in\\{0,1\\}.$

The group element $y$ is also a vector in $\mathbb{C}[\mathbb{Z}_{2}^{n}]$
whose matrix representation is
$\ket{y}=\ket{y_{1}}\otimes\cdots\otimes\ket{y_{n}}$. Hence, the action of
$\rho_{reg}(x)$ over $y$ can be rewritten as

$\begin{split}\rho_{reg}(x)y&=(X^{x_{1}}\ket{y_{1}})\otimes\cdots\otimes(X^{x_{n}}y_{n}),\mbox{
where }X^{x_{i}}\ket{y_{i}}=\ket{x_{i}\oplus y_{i}},\\\
&=\left(X^{x_{1}}\otimes\cdots\otimes
X^{x_{n}}\right)(\ket{y_{1}}\otimes\cdots\otimes\ket{y_{n}}).\end{split}$

The adjacency matrix $A$ of $Cay(\mathbb{Z}_{2}^{n},f)$ is decomposed by using
the regular representation on $\mathbb{Z}_{2}^{n}$, viz., given
$x,y\in\mathbb{Z}_{2}^{n}$, the value $\rho_{reg}(x)y=x\oplus y$ corresponds
to the $(x,y)$-entry of $A$, so $A$ can be expressed as;

$A=\sum_{x\in\mathbb{Z}_{2}^{n}}f(x)\rho_{reg}(x).$ (2.1)

Since $\rho_{reg}(x)$ commutes with $\rho_{reg}(y)$ for all
$x,y\in\mathbb{Z}_{2}^{n}$, the evolution operator $\mathcal{U}(t)=e^{-\iota
tA}$ is decomposed into;

$\mathcal{U}(t)=\prod_{x\in\mathbb{Z}_{2}^{n}}U(x,t),\qquad U(x,t)=e^{-\iota
tf(x)\rho_{reg}(x)}.$ (2.2)

###  2.3 PST or periodicity in weighted cubelike graphs

We simulate continuous-time quantum walk on $Cay(\mathbb{Z}_{2}^{n},f)$ and
verify the existence of perfect state transfer or periodicity as mentioned in
the following theorem.

######  Theorem 2.4.

[5, 15] Let $f:\mathbb{Z}_{2}^{n}\rightarrow\mathbb{Z}$ be an integer-valued
function. For $x\in\mathbb{Z}_{2}^{n}$, define a subset
$O_{x}=\\{y\in\mathbb{Z}_{2}^{n}:\braket{x}{y}\mbox{ mod }2=1\\}$. Let
$e_{i}$, $1\leq i\leq n$, denote the $n$-tuple with entry 1 at position $i$
and zero everywhere else. Let $\sigma\in\mathbb{Z}_{2}^{n}$ such that

$\sigma_{i}=1\mbox{ only if }\sum_{y\in O_{e_{i}}}f(y)\mbox{ mod }2=1.$ (2.3)

Then,

  1. 1.

if $\sigma$ is the identity element, i.e., $\sigma=\textbf{0}$, then
$Cay(\mathbb{Z}_{2}^{n},f)$ is periodic with period $\frac{\pi}{2}$,

  2. 2.

if $\sigma\neq\textbf{0}$, then PST occurs between every pair $\\{u,v\\}$
satisfying $u\oplus v=\sigma$, with time $\tau=\frac{\pi}{2}$.

######  Note.

Although PST or periodicity in weighted cubelike graph mentioned in [15] was
done independently, it was only later that the authors realized that its
generalized version, viz., PST on weighted abelian Cayley graph, has already
been proved in another paper [5].

##  3 The quantum simulation

The idea to design a quantum circuit for CTQW on a cubelike graph has been
taken from [17]; if the Hamiltonian is given by $A=Z_{1}\otimes\cdots\otimes
Z_{n}$, where $Z_{i}=Z$, then the phase shift applied to the system is
$e^{-\iota t}$ if the parity of the $n$ qubits in the computational basis is
even, otherwise, the phase shift applied is $e^{\iota t}$. Fig. 4 illustrates
the quantum circuit for $e^{-\iota tA}$, where $A=Z\otimes Z\otimes Z$.

$\Qcircuit<EMAIL_ADDRESS>\lstick{}\qw\qw\ctrl{2}\qw\qw\qw\ctrl{2}\qw\qw\qw\\\
\lstick{}\qw\qw\qw\ctrl{1}\qw\ctrl{1}\qw\qw\qw\qw\\\
\lstick{\ket{0}}\qw\targ\targ\targ\gate{e^{-\iota
tZ}}\targ\targ\targ\qw\qw\rstick{\ket{0}}}$ Figure 4: Quantum circuit to
implement $e^{-\iota tA}$, where $A=Z\otimes Z\otimes Z$.

###  3.1 Quantum circuits

$\Qcircuit@C=1em@R=1em{\lstick{\ket{x_{1}}}&\gate{\mathrm{H}}\qw\cdots\qw\qw\qw\cdots\qw\gate{\mathrm{H}}\qw\\\
\lstick{\vdots}\vdots\ddots\vdots\vdots\vdots\iddots\vdots\vdots\vdots\\\
\lstick{\ket{x_{i_{1}}}}\gate{\mathrm{H}}\ctrl{5}\cdots\qw\qw\qw\cdots\ctrl{5}\gate{\mathrm{H}}\qw\\\
\lstick{\vdots}\vdots\ddots\vdots\vdots\vdots\iddots\vdots\vdots\\\
\lstick{\ket{x_{i_{k}}}}\gate{\mathrm{H}}\qw\cdots\ctrl{3}\qw\ctrl{3}\cdots\qw\gate{\mathrm{H}}\qw\\\
\lstick{\vdots}\vdots\ddots\vdots\iddots\vdots\vdots\\\
\lstick{\ket{x_{n}}}\gate{\mathrm{H}}\qw\cdots\qw\qw\qw\cdots\qw\gate{\mathrm{H}}\qw\\\
\lstick{\ket{0}}\qw\targ\qw\qw\targ\gate{\mathrm{R_{\hat{z}}(2tf(x))}}\targ\qw\qw\targ\qw\qw}$
Figure 5: Quantum circuit for $U(x,t)=e^{-\iota tf(x)\rho_{reg}(x)}$.

Let $x\in\mathbb{Z}_{2}^{n}$, then the regular representation $\rho_{reg}(x)$
is given by

$\rho_{reg}(x)=\otimes_{i=1}^{n}X^{x_{i}}=H^{\otimes
n}\left(\otimes_{i=1}^{n}Z^{x_{i}}\right)H^{\otimes n},\mbox{ since }X=HZH.$

Applying the changes to the operator $U(x,t)$ in Eq. 2.2, we get

$\begin{split}U(x,t)&=e^{-\iota tf(x)\rho_{reg}(x)}=e^{-\iota
tf(x)\left[\otimes_{i=1}^{n}X^{x_{i}}\right]}\\\
&=\sum_{l=0}^{\infty}\frac{(-\iota
tf(x))^{l}}{l!}\left[\otimes_{i=1}^{n}X^{x_{i}}\right]^{l}\\\
&=\sum_{l=0}^{\infty}\frac{(-\iota tf(x)))^{2l}}{(2l)!}I^{\otimes
n}+\sum_{l=0}^{\infty}\frac{(-\iota
tf(x))^{2l+1}}{(2l+1)!}\left[\otimes_{i=1}^{n}X^{x_{i}}\right]\\\ &=H^{\otimes
n}V(x,t)H^{\otimes n},\qquad V(x,t)=e^{-\iota
tf(x)\left[\otimes_{i=1}^{n}Z^{x_{i}}\right]}.\end{split}$

We see that,

$\begin{split}\left(Z_{1}^{x_{1}}\otimes\cdots\otimes
Z_{n}^{x_{n}}\right)\ket{y}&=(-1)^{x_{1}y_{1}}\ket{y_{1}}\otimes\cdots\otimes(-1)^{x_{n}y_{n}}\ket{y_{n}}\\\
&=(-1)^{\sum_{i=1}^{n}x_{i}y_{i}}\ket{y_{1}}\otimes\cdots\otimes\ket{y_{n}}\\\
&=\begin{cases}\ket{y},&\mbox{if }\braket{x}{y}\mbox{ mod }2=0\\\
-\ket{y},&\mbox{if }\braket{x}{y}\mbox{ mod }2=1.\end{cases}\end{split}$

This implies,

$V(x,t)\ket{y}=\begin{cases}e^{-\iota tf(x)Z}\ket{y}&\mbox{ if
}\braket{x}{y}\mbox{ mod }2=0\\\ e^{\iota tf(x)Z}\ket{y}&\mbox{if
}\braket{x}{y}\mbox{ mod }2=1.\end{cases}$

Thus, the action of the operator $V(x,t)$ is equivalent to the application of
the rotation operator $R_{\hat{z}}(2tf(x))$ about the $\hat{z}$-axis if
$\braket{x}{y}$ is even, and $R_{\hat{z}}(-2tf(x))$ if $\braket{x}{y}$ is odd.
Hence, if $x$ has non-zero entries at positions $i_{1},\dots,i_{k}$, then the
quantum circuit for the operator $e^{-\iota tf(x)\rho_{reg}(x)}$ is depicted
by Fig. 5. Suppose elements in $\Omega_{f}=\\{y:f(y)\neq 0\\}$ are represented
by $\Omega_{f}=\\{x^{(1)},\dots,x^{(\Delta)}\\}$, where $\Delta$ is the
cardinality of $\Omega_{f}$, then the quantum circuit for the continuous-time
quantum walk is as shown in Fig. 6, where the initialized state, in general,
is $\ket{0}^{\otimes n}$ along with an ancilla qubit with state $\ket{0}$.

######  Remark 3.1.

As seen in Fig. 6, the Hadamard gates $H$ applied at the end of $U(x^{(i)},t)$
and the beginning of $U(x^{(i+1)},t)$, $1\leq i<\Delta$, are not required,
because $H^{2}=I$, thus the actual number of $H$ gates required are $2n$.
Secondly, the number of rotation operators used are $\Delta$. Lastly, for each
$x\in\Omega_{f}$, the number of CNOT gates applied are equal to the Hamming
weight $wt(x)$ of $x$. Thus, the total number of CNOT gates used are
$\sum_{x\in\Omega_{f}}wt(x)$.

$\Qcircuit@C=1em@R=1.5em{\lstick{q_{0}=\ket{0}}&\multigate{4}{\mathrm{U}(x^{(1)},t)}\cdots\multigate{4}{\mathrm{U}(x^{(\Delta)},t)}\meter\qw\qw\qw\qw\\\
\lstick{q_{1}=\ket{0}}\ghost{{\mathrm{U}(x^{(1)},t)}}\cdots\ghost{\mathrm{U}(x^{(\Delta)},t)}\qw\meter\qw\qw\qw\\\
\lstick{\vdots}\ghost{{\mathrm{U}(x^{(1)},t)}}\cdots\ghost{\mathrm{U}(x^{(\Delta)},t)}\ddots\\\
\lstick{q_{n-1}=\ket{0}}\ghost{{\mathrm{U}(x^{(1)},t)}}\cdots\ghost{\mathrm{U}(x^{(\Delta)},t)}\qw\qw\qw\meter\qw\qw\\\
\lstick{\ket{0}}\ghost{{\mathrm{U}(x^{(1)},t)}}\cdots\ghost{\mathrm{U}(x^{(\Delta)},t)}\qw\qw\qw\qw\qw\qw\\\
\lstick{c:}\lstick{/_{{}_{{}_{2}}}}\cw\cw\cw\cw\dstick{{}_{{}_{0}}}\cwx[-5]\cw\dstick{{}_{{}_{1}}}\cw\cwx[-4]\cw\dstick{{}_{{}_{n-1}}}\cw\cwx[-2]\cw\cw}$
Figure 6: An illustration of CTQW quantum circuit on weighted cubelike graph

###  3.2 Results

Recall that, if $u\oplus v=\sigma$, where $\sigma$ is given by Eq. 2.3 in
Theorem 2.4, then $\\{u,v\\}$ is the PST pair. This partitions the vertex set
into PST pairs. The graph shown in Fig. 2 admits PST between pairs
$\\{000,111\\}$, $\\{001,110\\}$, $\\{010,101\\}$, $\\{011,100\\}$, and the
other graph in Fig. 2 has PST pairs $\\{000,011\\}$, $\\{001,010\\}$,
$\\{100,111\\}$, $\\{101,110\\}$. Since weighted cubelike graphs, as described
in Theorem 2.4, are vertex-transitive, the study of PST between the pair
$\\{\textbf{0},\sigma\\}$ is equivalent to any other pair. Therefore, every
quantum circuit is initialized to state $\ket{0}^{\otimes n}$, see Fig. 7 and
Fig. 8 which illustrate quantum circuits for the above graphs mentioned.

Suppose the weight function $f$ is defined by

$f(001)=4,\;f(011)=8,\;\mbox{and },\;f(101)=3,$ (3.1)

and zero on other elements, then the $3$-tuple $\sigma$ is computed as (using
Theorem 2.4);

$\begin{split}O_{001}=\\{001,011,101\\}&\implies f(001)+f(011)+f(101)\mbox{
mod }2=1\\\ &\implies\sigma_{1}=1\\\ O_{010}=\\{011\\}&\implies f(011)\mbox{
mod }2=0\\\ &\implies\sigma_{2}=0\\\ O_{100}=\\{101\\}&\implies f(101)\mbox{
mod }2=1\\\ &\implies\sigma_{3}=1\end{split}$

Thus, $\sigma=101$ and $\\{000,101\\}$ is a PST pair. The same is obtained by
simulating the quantum circuit shown in Fig. 9.

Figure 7: Quantum circuit for $Cay(\mathbb{Z}_{2}^{3},\\{001,010,100\\})$.

Figure 8: Quantum circuit for
$Cay(\mathbb{Z}_{2}^{3},\\{001,010,011,100,111\\})$.

Figure 9: Quantum circuit for
$Cay(\mathbb{Z}_{2}^{3},\\{f(001)=4,f(011)=8,f(101)=3\\})$.

On the other hand, if $f$ is defined by

$f(010)=4,\;f(011)=7,\;f(100)=8,\;f(101)=2,\;f(110)=5,$ (3.2)

then $\sigma=101$, and $\\{000,101\\}$ is a PST pair.

######  Remark 3.2.

Given a pair in a cubelike graph, we can assign weights to edges such that PST
occurs between the given pair.

######  Note.

Quantum circuits displayed in Fig. 6 can not be run on real quantum computers
due to some techincal issues such as quantum decoherence and state fidelity.
We have, however, tested small graphs on the computer ibmq_manila as shown in
Fig. 10.

Figure 10: Experimented probability distribution of CTQW on
$Cay(\mathbb{Z}_{2}^{3},\\{01,10\\})$ (left) and on
$Cay(\mathbb{Z}_{2}^{3},\\{001,010,100\\})$ (right) after time
$\frac{\pi}{2}$.

##  4 Conclusion and future work

In this paper, we have experimentally tested perfect state transfer on IBM’s
quantum simulators and quantum computers on weighted cubelike graphs. We have
used Hamiltonian simulation techniques to construct efficient circuits for
continuous-time quantum random walks. We have verified the theoretical results
of [5] and [15] that PST or periodicity on integral weighted cubelike graphs
occurs at time $t=\frac{\pi}{2}$, where weights are determined by Theorem 2.4.
In the future, we plan to construct efficient quantum circuits for quantum
walks on weighted abelian Cayley graphs.

## References

  * [1] Abraham, H., and et. al. Qiskit: An open-source framework for quantum computing, 2019.
  * [2] Acasiete, F., Agostini, F., Moqadam, J., and Portugal, R. Implementation of quantum walks on ibm quantum computers.  Quantum Inf Process 19, 426 (2020).
  * [3] Aharonov, Y., Davidovich, L., and Zagury, N. Quantum random walks.  Phys. Rev. A 48 (Aug 1993), 1687–1690.
  * [4] Bernasconi, A., Godsil, C., and Severini, P. Quantum networks on cubelike graphs.  Phys. Rev. A 78, 052320 (2008).
  * [5] Cao, X., Feng, K., and Tan, Y.-Y. Perfect state transfer on weighted abelian cayley graphs.  Chinese Annals of Mathematics, Series B 42, 4 (2021), 625–642.
  * [6] Cheung, W., and Godsil, C. Perfect state transfer in cubelike graphs.  Linear Algebra and its Applications 435 (2011), 2468–2474.
  * [7] Farhi, E., and Gutmann, S. Quantum computation and decision trees.  Phys. Rev. A 58, 2 (1998), 915–928.
  * [8] Godsil, C. Periodic graphs.  The Electronic Journal of Combinatorics 18, 23 (01 2011).
  * [9] Godsil, C. State transfer on graphs.  Discrete Mathematics 312(1) (2012), 129–147.
  * [10] IBM.  Quantum.  https://quantum-computing.ibm.com/ (2021).
  * [11] J., A., Adedoyin, A., and et al. Quantum algorithm implementations for beginners.  arXiv:1804.03719 (2020).
  * [12] Kempe, J. Quantum random walks: An introductory overview.  Contemporary Physics 44, 4 (2003), 307–327.
  * [13] Kempe, J. Discrete quantum walks hit exponentially faster.  Probability Theory and Related Fields 133 (2005), 215–235.
  * [14] Mulherkar, J., Rajdeepak, R., and Sunitha, V. Implementation of hitting times of discrete time quantum random walks on cubelike graphs.  arXiv:2108.13769 (2021).
  * [15] Mulherkar, J., Rajdeepak, R., and Sunitha, V. Perfect state transfer in weighted cubelike graphs.  arXiv:2109.12607 (2021).
  * [16] Nayak, A., and Vishwanath, A. Quantum walk on the line.  arXiv:quant-ph/0010117 (2000).
  * [17] Nielsen, M. A., and Chuang, I. L. Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th ed.  Cambridge University Press, USA, 2011.
  * [18] Portugal, R. Quantum Walks and Search Algorithms, 2 ed.  Springer Publishing Company, Incorporated, Switzerland, 2013.
  * [19] Venegas-Andraca, S. E. Quantum walks: a comprehensive review.  Quantum Information Processing 11, 5 (2012), 1015–1016.
  * [20] Wanzambi, E., and Andersson, S. Quantum computing: Implementing hitting time for coined quantum walks on regular graphs.  arXiv:2108.02723 (2021).
  * [21] Warat, P., Prapong, P., and Unchalisa, T. Implementation of quantum random walk on a real quantum computer.  Journal of Physics: Conference Series 1719 (2021), 012103.