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A subgroup  X  of a group  G  is called  transitively normal  if  X  is normal in any subgroup  Y  of  G  such that    $$X\le Y$$  X ≤ Y and  X  is subnormal in  Y . Thus all subgroups of a group  G  are transitively normal if and only if normality is a transitive relation in every subgroup of  G  (i.e.  G  is a    $$\overline{T}$$  T ¯ -group). It is proved that a group  G  with no infinite simple sections satisfies the minimal condition on subgroups that are not transitively normal if and only if either  G  is Černikov or a    $${\overline{T}}$$  T ¯ -group.

Groups satisfying the minimal condition on subgroups which are not transitively normal