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Algebraic and logical methods in computer science and artificial intelligence
On two properties of Shunkov group
A. A. Shlepkin,
I. V. Sabodakh Siberian Federal University, Krasnoyarsk, Russian Federation
Abstract:
One of the interesting classes of mixed groups ( i.e. groups that can contain both elements of finite order and elements of infinite order) is the class of Shunkov groups. The group
$G$ is called Shunkov group if for any finite subgroup
$H$ of
$G$ in the quotient group
$N_G(H)/H$, any two conjugate elements of prime order generate a finite group. When studying the Shunkov group
$G$, a situation often arises when it is necessary to move to the quotient group of the group
$G$ by some of its normal subgroup
$N$. In which cases is the resulting quotient group
$G/N$ again a Shunkov group? The paper gives a positive answer to this question, provided that the normal subgroup
$N$ is locally finite and the orders of elements of the subgroup
$N$ are mutually simple with the orders of elements of the quotient group
$G/N$.
Let
$ \mathfrak{X}$ be a set of groups. A group
$G$ is saturated with groups from the set
$ \mathfrak{X}$ if any finite subgroup of
$G$ is contained in a subgroup of
$ G$ that is isomorphic to some group of
$\mathfrak{X}$ . If all elements of finite orders from the group
$G$ are contained in a periodic subgroup of the group
$G$, then it is called the periodic part of the group
$G$ and is denoted by
$T(G)$. It is proved that the Shunkov group saturated with finite linear and unitary groups of degree 3 over finite fields has a periodic part that is isomorphic to either a linear or unitary group of degree 3 on a suitable locally finite field.
Keywords:
Shunkov group, groups saturated with a given set of groups, periodic part of group.
UDC:
512.54
MSC: 20E25 Received: 23.01.2021
DOI:
10.26516/1997-7670.2021.35.103
Fulltext:
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