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Groups satisfying the minimal condition on non-pronormal subgroups. (English) Zbl 0837.20040
A subgroup $$H$$ of a group $$G$$ is said to be pronormal if for every element $$x \in G$$ the subgroups $$H$$ and $$H^x$$ are conjugate in $$\langle H, H^x\rangle$$.
The main result of this paper is Theorem. Let $$G$$ be a group satisfying the minimal condition on non-pronormal subgroups. If $$G$$ has no infinite simple sections (in particular, if $$G$$ is a locally graded group) then either $$G$$ is a Chernikov group or every subgroup of $$G$$ is pronormal. In particular, if $$G$$ is not torsion then $$G$$ is abelian.

##### MSC:
 2e+16 Chains and lattices of subgroups, subnormal subgroups 2e+08 Subgroup theorems; subgroup growth 2e+35 General structure theorems for groups