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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.
##### MSC:
 2e+16 Chains and lattices of subgroups, subnormal subgroups 2e+08 Subgroup theorems; subgroup growth