Structure of supercopiable p-groups (Q1823310)

The author investigated finite groups admitting a given system of endomorphisms [see: Mat. Zap. 14, No.1, 85-100 (1985; Zbl 0647.20012)]. This paper continues the investigation. A finite group is said to be copiable (quasicopiable) if each of its normal (minimal normal) subgroups is the kernel of an endomorphism of the entire group. A finite group G is said to be supercopiable if all of its quotient groups are copiable. As follows from the author's earlier investigation, in order to complete the classification of supercopiable p-groups it remains to study supercopiable p-groups in which the centers of all quotient groups are elementary, while the center of the group itself lies in the commutator subgroup and has prime order. In the present paper, the author describes the properties of a supercopiable p-group G whose center intersects the commutator group in a subgroup Z of prime order. Some of the properties are related to an endomorphism of G with the kernel Z.