On Baer's problem and properties of $M''$-groups

Abstract

We establish many new properties of $M''$-groups and give a large set of new counter-examples to the well-known problems of R. Baer (1949) and S. N. Chernikov (1959) concerning socle groups, $M''$-groups and $M'$-groups. In passing we also show that for a periodic $FC$-group $G$ and its locally soluble radical $R$ the factor group $G/L(G)R$ is residually finite, where $L(G)$ is the product of all normal semisimple subgroups of $G$.