Characterization of generalized Chernikov groups among groups with involutions

The class of generalized Chernikov groups is characterized, i.e., the class of periodic locally solvable groups with the primary ascending chain condition. The name of the class is related to the fact that the structure of such groups is close to that of Chernikov groups. Namely, a Chernikov group is defined as a finite extension of a direct product of finitely many quasi-cyclic groups, and a generalized Chernikov group is a layer-finite extension of a direct product $A$ of quasi-cyclic $p$-groups with finitely many factors for each prime $p$ such that each of its elements does not commute elementwise with only finitely many Sylow subgroups of $A$. A theorem that characterizes the generalized Chernikov groups in the class of groups with involution is proved.