Permutable implies ascendant

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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., ascendant subgroup)
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Any permutable subgroup of a group is an ascendant subgroup.

Related facts

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Facts used

  1. Permutable subgroup is normalized by any trivially intersecting infinite cyclic group


Textbook references

  • Subnormal subgroups of groups by John C. Lennox and Stewart E. Stonehewer, Oxford Mathematical Monographs, ISBN 019853552X, Page 215, Theorem 7.1.6, More info