The normalizer \(N_G(H)\) consists of the elements of \(G\) that preserve \(H\) under conjugation. It is the largest subgroup of \(G\) in which \(H\) is normal and converts the embedding of a local subgroup into a computable stabilizer. [@isaacs2008]
Definition
The convention is \(H^g=g^{-1}Hg\). The normalizer of a cyclic subgroup may send a generator to another generator or power, while a centralizer must fix the generator itself.
Core result
The stabilizer condition is exactly \(H^g=H\), and orbit-stabilizer gives the index. The Frattini argument \(G=N_G(P)N\) for \(P\in\operatorname{Syl}_p(N)\) is proved in the Sylow core article and Isaacs, Theorem 1.13.
Structural properties
- \(H\triangleleft G\) exactly when \(N_G(H)=G\); \(H\) is self-normalizing exactly when \(N_G(H)=H\).
- If \(P\) is a finite \(p\)-group and \(H<P\), then \(H<N_P(H)\), the normalizer-growth property.
- A finite group is nilpotent exactly when every proper subgroup is properly contained in its normalizer.
- If \(P\in\operatorname{Syl}_p(G)\), every \(p\)-subgroup of \(N_G(P)\) lies in \(P\).
- The Sylow normalizer is a natural candidate for controlling fusion and transfer; Yoshida's theorem gives broad conditions for transfer control.
The normalizer is the largest ambient subgroup in which one may use conjugation without destroying the chosen subgroup, making it the standard bridge from local to global structure.
Example and boundary
Knowledge network
The Sylow article uses normalizer counts and the Frattini argument; the nilpotent-group article promotes normalizer growth to an equivalent characterization.
Proof and sources
Conjugacy counting, Sylow normalizers, and normalizer growth are Isaacs, Theorems 1.6, 1.18, and 1.22; proofs are linked to the corresponding core articles. [@isaacs2008]
