On periodic groups of Shunkov with the Chernikov centralizers of involutions

Layer-finite groups first appeared in the work by S. N. Chernikov (1945). Almost layer-finite groups are extensions of layer-finite groups by finite groups. The author develops the direction of characterizing the well studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. In this paper, almost layer-finite groups are characterized in the class of periodic Shunkov groups. Shunkov group is a group $G$ in which for any of its finite subgroup $ K $ in the factor group $N_G (K) / K$ any two conjugate elements of prime order generate a finite subgroup. We study periodic Shunkov groups under the condition that a normalizer of any finite nontrivial subgroup is almost layer-finite. It is proved that if in such a group the centralizers of involutions are Chernikov ones, then the group is almost layer-finite.