Sylow-permutable implies subnormal in finite

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Sylow-permutable subgroup (?)) must also satisfy the second subgroup property (i.e., Subnormal subgroup (?)). In other words, every Sylow-permutable subgroup of finite group is a subnormal subgroup of finite group.
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Statement

In a finite group, any Sylow-permutable subgroup (i.e., any subgroup that permutes with all Sylow subgroups) is subnormal.

Related facts

• Conjugate-permutable implies subnormal in finite

Facts used

1. Sylow-permutability satisfies intermediate subgroup condition
2. Maximal Sylow-permutable implies normal