Sylow normalizer implies weakly abnormal
From Groupprops
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 normalizer (?)) must also satisfy the second subgroup property (i.e., Upward-closed self-normalizing subgroup (?)). In other words, every Sylow normalizer of finite group is a upward-closed self-normalizing subgroup of finite group.
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Statement
The normalizer of any Sylow subgroup in a finite group is a weakly abnormal subgroup: any subgroup containing it is a Self-normalizing subgroup (?).
Facts used
Proof
The proof follows directly by combining facts (1) and (2).
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 147, Exercise 32, Section 4.5 (Sylow's theorem)