# zbMATH — the first resource for mathematics

Groups with pronormal subgroups of proper normal subgroups. (English) Zbl 0866.20028
de Giovanni, Francesco (ed.) et al., Infinite groups 1994. Proceedings of the international conference, Ravello, Italy, May 23-27, 1994. Berlin: Walter de Gruyter. 247-256 (1996).
A subgroup $$H$$ of a group $$G$$ is said to be pronormal if for every element $$x$$ of $$G$$ the subgroups $$H$$ and $$H^x$$ are conjugate in $$\langle H,H^x\rangle$$. In this paper the author considers groups in which all subgroups of each proper normal subgroup are pronormal, and he obtains a complete description of such groups in the locally soluble case. Moreover, he describes locally soluble groups in which all non-normal cyclic primary subgroups are abnormal. Recall here that a subgroup $$H$$ of a group $$G$$ is abnormal if $$x$$ belongs to $$\langle H,H^x\rangle$$ for every $$x$$ in $$G$$.
For the entire collection see [Zbl 0856.00019].
##### MSC:
 20F16 Solvable groups, supersolvable groups 20E15 Chains and lattices of subgroups, subnormal subgroups 20F19 Generalizations of solvable and nilpotent groups