On Direct Products of Dihedral Groups in Locally Finite Groups

When studying infinite groups, as a rule, some finiteness conditions are imposed. For example, they require that the group be periodic, a Shunkov group, a Frobenius group, or a locally finite group. The concept of saturation allows us to effectively establish the internal structure of various classes of infinite groups. To date, a large array of results on groups saturated with various classes of finite groups has been obtained. Another important direction in the study of groups with saturation conditions is the study of groups saturated by direct products of various groups. Significant progress has been made in solving the problem of B. Amberg and L. S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. It is proved that a locally finite group saturated by the direct product of a finite number of finite groups of a dihedron is isomorphic to the direct product of locally cyclic groups multiplied by an involution. It is also proved that a locally finite group saturated by the direct product of a finite number of finite dihedral groups is solvable

Regular permutations and their applications in crystallography

The representation of a group G in the form of regular permutations is widely used for studying the structure of finite groups, in particular, parameters like the group density function. This is related to the increased potential of computer technologies for conducting calculations. The work addresses the problem of calculation regular permutations with restrictions on the structure of the degree and order of permutations. The considered regular permutations have the same nontrivial order, which divides the degree of the permutation. Examples of the application of permutation groups in crystallography and crystal chemistry are provided

Об одном достаточном условии, при котором бесконечная группа не будет простой

We describe the conditions of existing periodic part in Shunkov groupВ работе рассмотрены условия существования периодической части группы Шунков

Об одном достаточном условии, при котором бесконечная группа не будет простой

We describe the conditions of existing periodic part in Shunkov groupВ работе рассмотрены условия существования периодической части группы Шунков