On Two Properties of Shunkov Group

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. \end{abstracte}