Permutable implies ascendant

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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Statement

Any permutable subgroup of a group is an ascendant subgroup.

Related facts

Similar facts

• Permutable implies subnormal in finite
• Permutable implies locally subnormal

Opposite facts

• Permutable not implies subnormal
• There exist permutable subgroups of arbitrarily large subnormal depth

Facts used

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

References

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}