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Groups with only abnormal and subnormal subgroups. (English) Zbl 0918.20017
A subgroup $$H$$ of a group $$G$$ is said to be abnormal in $$G$$ if $$g$$ belongs to the subgroup $$\langle H,H^g\rangle$$ for every element $$g$$ of $$G$$. In this paper the authors give a complete description of non-perfect groups in which every subgroup either is subnormal or abnormal. A subgroup $$H$$ of a group $$G$$ is said to be counternormal if the normal closure $$H^G$$ of $$H$$ in $$G$$ coincides with $$G$$. Every abnormal subgroup is also counternormal, while the converse is not true. The authors characterize also non-perfect groups in which every subgroup either is subnormal or counternormal, and this last theorem generalizes a previous result of I. Ya. Subbotin [Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 3(358), 86-88 (1992; Zbl 0816.20030)].

##### MSC:
 20E15 Chains and lattices of subgroups, subnormal subgroups 20F19 Generalizations of solvable and nilpotent groups