Restoration of information on the group by the bottom layer

The question of the possibility of restoring information on the group by its bottom layer is considered. The problem is classical for mathematical modeling: restoration of missing information on the object employing part of the saved data. This problem will be solved in the class of layer-finite groups. A group is said to be layer-finite if it has a finite number of elements of every order. This concept was first introduced by S. N. Chernikov. It appeared in connection with the study of infinite locally finite p-groups in the case when the center of the group has a finite index in it. The bottom layer of the group G is the set of its prime order elements. By the bottom layer of the group, you can sometimes restore the group or judge about the properties of such a group. Among these results one can name those that completely describe the structure of the group by its bottom layer, for example: if the bottom layer of the group G consists of elements of order 2 and there are no non-unit elements of other orders in the group, then G is the elementary Abelian 2-group. V. P. Shunkov proved that if the bottom layer in an infinite layer-finite group consists of one element of order 2, then the group G is either a quasicyclic or an infinite generalized quaternion group. We will restore the information on the group by its bottom layer. This problem will be solved in the class of layer-finite groups. Group G is said to be recognizable by the bottom layer if it is uniquely recovered by the bottom layer. Group G is said to be almost recognizable over the bottom layer if there is a finite number of pairwise nonisomorphic groups with the same bottom layer as in group G. Group G is said to be unrecognizable by the bottom layer if there is an infinite number of pairwise nonisomorphic groups with the same bottom layer such as in group G. In this work conditions under which the group is recognized align the bottom layer have been established.