
This article is cited in 4 scientific papers (total in 4 papers)
Some residual properties of soluble groups of finite rank
D. N. Azarov^{} ^{} Ivanovo State University
Abstract:
The generalization of one
classical Smel'kin's theorem
for polycyclic groups is obtained.
A. L. Smelkin proved that if $G$ is
a polycyclic group,
then it is a virtually
residually finite $p$group
for any prime $p$.
Recall that a group $G$ is said to
be a residually finite $p$group if
for every nonidentity element $a$
of $G$ there exists a homomorphism
of the group $G$ onto some finite
$p$group such that the image of
the element $a$ differs from 1.
A group $G$ will be said to be a virtually
residually finite $p$group if it contains
a finite index subgroup
which is a residually finite $p$group.
One of the generalizations of the
notation of polycyclic group is a
notation of soluble finite rank group.
Recall that a group $G$ is said to
be a group of finite rank if there
exists a positive integer $r$ such
that every finitely generated
subgroup in $G$ is generated by
at most $r$ elements.
For soluble groups of finite rank
the following
necessary and sufficient condition
to be a residually finite
$\pi $group for some finite set
$\pi $ of primes is obtained.
If $G$ is a group of finite rank,
then the group $G$ is a
residually finite $\pi $group
for some finite set $\pi $ of primes
if and only if $G$ is a reduced
poly(cyclic, quasicyclic, or
rational) group.
Recall that a group $G$ is said to
be a reduced group if it has no
nonidentity radicable subgroups.
A group $H$ is said to be a
radicable group if every element
$h$ in $H$ is an $m$th power of an
element of $H$ for every
positive number $m$.
It is proved that if a soluble group of finite rank
is a residually finite
$\pi $group for some finite set
$\pi $ of primes, then it is
a virtually residually finite nilpotent $\pi $group.
We prove also the following
generalization of Smel'kin's theorem.
Let $\pi $ be a finite set of
primes. If $G$ is a soluble
group of finite rank,
then the group $G$ is a virtually
residually finite $\pi $group
if and only if $G$ is a reduced
poly(cyclic, quasicyclic, or
rational) group and $G$ has no
$\pi $radicable elements
of infinite order.
Recall that an element $g$ in $G$
is said to be $\pi $radicable
if $g$ is an $m$th power of an
element of $G$ for every
positive $\pi $number $m$.
Keywords:
finite rank group, soluble group, polycyclic group, nilpotent group, residually finite $p$group.
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UDC:
512.543 Received: 31.01.2014
Citation:
D. N. Azarov, “Some residual properties of soluble groups of finite rank”, Chebyshevskii Sb., 15:1 (2014), 7–18
Citation in format AMSBIB
\Bibitem{Aza14}
\by D.~N.~Azarov
\paper Some residual properties of soluble groups of finite rank
\jour Chebyshevskii Sb.
\yr 2014
\vol 15
\issue 1
\pages 718
\mathnet{http://mi.mathnet.ru/cheb321}
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http://mi.mathnet.ru/eng/cheb321 http://mi.mathnet.ru/eng/cheb/v15/i1/p7
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This publication is cited in the following articles:

D. N. Azarov, “Approksimatsionnye svoistva nilpotentnykh grupp”, Model. i analiz inform. sistem, 22:2 (2015), 149–157

D. N. Azarov, “Approksimatsionnye svoistva abelevykh grupp”, Vestn. Tomsk. gos. unta. Matem. i mekh., 2015, no. 3(35), 5–11

D. V. Goltsov, “Approksimiruemost fundamentalnoi gruppy konechnogo grafa grupp kornevym klassom grupp”, Chebyshevskii sb., 17:3 (2016), 64–71

D. N. Azarov, N. S. Romanovskii, “Finite homomorphic images of groups of finite rank”, Siberian Math. J., 60:3 (2019), 373–376

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