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

Abstract

AbstractA 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.