Sylow-permutable implies subnormal in finite

From Groupprops
Jump to: navigation, search
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.
View all subgroup property implications in finite groups | View all subgroup property non-implications in finite groups | View all subgroup property implications | View all subgroup property non-implications

Statement

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

Related facts

Facts used

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