Let \(x\in G\) and \(x^G\) be its conjugacy class. Then \(x\) is called an MC-element if \(G/C_G(x^G)\) is a (soluble minimax)-by-finite group; further \(M(G)\) is the subgroup of all MC-elements of \(G\). The group \(G\) is MC-hypercentral if the sequence \(M_1=M(G)\), \(M_{i+1}/M_i=M(G/M_i)\) ends in \(G\) (possibly transfinitely).
The author describes MC-hypercentral groups with maximum (minimum) condition for normal subgroups. Also he takes a closer look at these groups with finite Abelian section rank, and shows that MC-hypercentrality coincides with other properties of this kind like FC-hypercentrality.