Permutable subgroup is normalized by any trivially intersecting infinite cyclic group

Statement

Suppose $H$ is a Permutable subgroup (?) of a group $G$ and $K$ is an infinite cyclic subgroup of $G$ that intersects $H$ trivially. Then, $K$ normalizes $H$: in other words, $K$ is contained in the normalizer $N_{G}(H)$.

References

Textbook references

• Subnormal subgroups of groups by John C. Lennox and Stewart E. Stonehewer, Oxford Mathematical Monographs, ISBN 019853552X, Page 215, Lemma 7.1.7, ^{More info}