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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:
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
20E34 General structure theorems for groups
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