Groups with only abnormal and subnormal subgroups. (English) Zbl 0918.20017

De Falco, Maria; Kurdachenko, Leonid A.; Subbotin, Igor Ya.

Atti Semin. Mat. Fis. Univ. Modena 46, No. 2, 435-442 (1998).

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)].

Reviewer: S.Franciosi (Napoli)

MSC:

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

Keywords:

abnormal subgroups; subnormal subgroups; counternormal subgroups

Citations:

Zbl 0816.20030