On groups whose transitively normal subgroups are either normal or self-normalizing. (Q380400)

A subgroup \(M\) of a group \(G\) is transitively normal if \(M\) is a normal subgroup of any subgroup \(T\) in which it is subnormal. Soluble T-groups and many locally nilpotent groups belong to the class of groups indicated in the title: The intersection of this class with the classes of hyperfinite groups, locally finite hypofinite groups and periodic soluble groups that are Sylow \(p\)-integrated for all primes \(p\) are periodic locally-nilpotent-by-finite groups (Theorem A). Periodic locally-nilpotent-by-finite groups \(G\) of the title are either locally nilpotent or generated by an element \(x\) and a locally nilpotent normal subgroup \(N\) such that \(x^p\in N\), \(\langle x^G\rangle=G\), and \(C(x)\) is a \(p\)-group, where \(p\) is a prime. In particular, the Hall \(p'\)-subgroup of \(N\) is nilpotent, and if \(G\) is finite, then \(\langle x\rangle\) is a Sylow \(p\) subgroup of \(G\) (Theorem B and C). Also CC-groups, MC-groups and soluble-by-finite minimax groups are described.