On periodic groups of Shunkov with the Chernikov centralizers of involutions

Abstract

Layer-finite groups for the first time appeared in the article of S.~N.~Chernikov (1945). Almost layer-finite groups are extensions of a layer-finite groups by finite groups. The author develops the direction of characterizations of known well studied classes of groups in other classes of groups with some additional (rather weak) finiteness conditions. In this paper almost layer-finite groups receive characterization 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's groups with the condition: normalizer of any finite non-unit subgroup is almost layer-finite. It is proved that if in such group the centralizers of involutions are Chernikov, then the group is almost layer-finite.