The centralizer \(C_G(X)\) records the part of the ambient group that commutes elementwise with \(X\). It is both a stabilizer for conjugation and a basic local measure of noncommutativity. [@isaacs2008]
Definition
For \(H\leq G\), the centralizer \(C_G(H)\) is normal in \(N_G(H)\), and conjugation embeds \(N_G(H)/C_G(H)\) faithfully in \(\operatorname{Aut}(H)\).
Core result
Under conjugation, \(x^g=x\) exactly when \(gx=xg\), so the stabilizer is \(C_G(x)\) and orbit-stabilizer gives the class size. The normalizer quotient embeds through the conjugation homomorphism.
Structural properties
- \(C_G(X)=\bigcap_{x\in X}C_G(x)\) and is therefore a subgroup.
- It depends only on \(\langle X\rangle\); a generating set is enough to compute the centralizer of a subgroup.
- If \(M,N\triangleleft G\) and \(M\cap N=1\), then \([M,N]\leq M\cap N=1\), so they centralize one another.
- In a finite group, \(|x^G|=|G:C_G(x)|\): large centralizers correspond to small conjugacy classes.
- The generalized Fitting theorem \(C_G(F^*(G))\leq F^*(G)\) is a central example of global control through a centralizer.
A normalizer preserves a subgroup setwise, while a centralizer fixes it elementwise. Their quotient is exactly the automorphism group induced by conjugation.
Example and boundary
Knowledge network
The class equation counts element centralizers, the commutator article reformulates mutual centralization, and generalized Fitting theory supplies a canonical self-centralizing layer.
Proof and sources
The class formula is Isaacs, Theorem 1.5. Trivially intersecting normal subgroups centralize each other in Problem 1F.1 and Theorem 2.7; commutator language is developed in Section 4A. [@isaacs2008]
