## 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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### References:

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