×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite