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}

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Computational group theory investigates the arithmetic properties of algebraic structures. These properties may be utilized in development of algorithms for data analysis and machine learning. This direction is associated with the design, analysis of algorithms and data structures for calculating various characteristics for (mostly finite) groups. The area is interesting for investigation of important from various points of view groups that are important, the data about which cannot be obtained by manually. The present work was carried out in line with computer calculations in groups. Based on the concept of the group growth function, the concept of the group density function is introduced. The growth and density functions of finite simple non-Abelian groups of small orders are constructed. The question of recognizability of finite simple non-Abelian groups by the density function is considered. The problem of effective storage of finite group elements in computer memory using the AVL tree is investigated. Based on the AVL tree, a fast algorithm for enumerating all elements of a finite group is developed. With the computer calculations, the recognizability of two finite simple non-Abelian groups by the group density function is proved.