A generalization of groups with many almost normal subgroups

A subgroup H of a group G is called almost normal in G if it has finitely many conjugates in G. A classic result of B. H. Neumann informs us that |G:Z(G)| is finite if and only if each H is almost normal in G. Starting from this result, we investigate the structure of a group in which each non-finitely generated subgroup satisfies a property, which is weaker to be almost norma

Anti-PCPC-groups and Anti-CCCC-groups

A group $G$ has Chernikov classes of conjugate subgroups if the quotient group $G/ core_G(N_G(H))$ is a Chernikov group for each subgroup $H$ of $G$. An anti-$CC$-group $G$ is a group in which each nonfinitely generated subgroup $K$ has the quotient group $G/core_G(N_G(K))$ which is a Chernikov group. Analogously, a group $G$ has polycyclic-by-finite classes of conjugate subgroups if the quotient group $G/core_G(N_G(H))$ is a polycyclic-by-finite group for each subgroup $H$ of $G$. An anti-$PC$-group $G$ is a group in which each nonfinitely generated subgroup K has the quotient group $G/core_G(N_G(K))$ which is a polycyclic-by-finite group. Anti-$CC$-groups and anti-$PC$-groups are the subject of the present article