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Groups with all subgroups either subnormal or self-normalizing. (English) Zbl 1078.20026

Summary: Let \(G\) be a group in which every subgroup that is not self-normalizing is subnormal. It is shown that if \(G\) is locally finite then either \(G\) has a nilpotent subgroup of prime index or every subgroup of \(G\) is subnormal, while if \(G\) is not periodic then every subgroup of \(G\) is subnormal. If \(G\) is a periodic group with the given property then \(G\) need not be locally finite, but if \(G\) is also locally graded then it is locally finite.

MSC:

20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
20F50 Periodic groups; locally finite groups
20F18 Nilpotent groups
20E25 Local properties of groups
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