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Polynormal subgroups. (English) Zbl 0807.20026
Let \(H\) be a subgroup of a group \(G\). A system \(\{H_ i \mid i \in I\}\) of subgroups of \(G\) containing \(H\) is called a fan for \(H\) if for each subgroup \(K\) of \(G\) containing \(H\) there exists a unique index \(i \in I\) such that \(H_ i \leq K \leq N_ G(H_ i)\). A subgroup \(H\) of \(G\) is said to be fan subgroup if there exists a fan for \(H\). A subgroup \(F\) of \(G\) containing \(H\) is said to be full (relatively to \(H\)) if \(F\) coincides with the normal closure \(H^ F\) of \(H\) in \(F\). Finally, a subgroup \(H\) of \(G\) is said to be polynormal if the set of all full subgroups of \(G\), relatively to \(H\), is the fan for \(H\) in \(G\). In this paper the author investigates properties of polynormal subgroups of a group.
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth