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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].
20F16 Solvable groups, supersolvable groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20F19 Generalizations of solvable and nilpotent groups