Permutable implies locally subnormal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., permutable subgroup) must also satisfy the second subgroup property (i.e., locally subnormal subgroup)
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Statement

Any permutable subgroup of a group is a locally subnormal subgroup -- it is subnormal in its join with any finitely generated subgroup.

Related facts

References

Textbook references

  • Subnormal subgroups of groups by John C. Lennox and Stewart E. Stonehewer, Oxford Mathematical Monographs, ISBN 019853552X, Page 216, Theorem 7.1.8, More info