Locally finite groups saturated with direct product of two finite dihedral groups

In the study of infinite groups, as a rule, some finiteness conditions are imposed. For example, the group is required to be periodic, Shunkov group, Frobenius group, locally finite group. The concept of saturation makes it possible 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 with direct products of various groups. In this paper, a partial solution to the problem of B. Amberg and L.S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. The structure of a locally finite group saturated with a direct product of two finite dihedral groups is established and it is proved that in this case the group is solvable. The result obtained is an important step towards solving the problem of Amberg and Kazarin.