# centralizer

Let $G$ a group acting on itself by conjugation. Let $X$ be a subset of $G$. The stabilizer of $X$ is called the centralizer of $X$ and it’s the set

 $C_{G}(X)=\{g\in G:gxg^{-1}=x\quad\mbox{for all }x\in X\}$

For any group $G$, $C_{G}(G)=Z(G)$, the center of $G$. Thus, any subgroup of $C_{G}(G)$ is an abelian subgroup of $G$. However, the converse is generally not true. For example, take any non-abelian group and pick any element not in the center. Then the subgroup generated by it is obviously abelian, clearly non-trivial and not contained in the center.

Title centralizer Centralizer1 2013-03-22 14:01:20 2013-03-22 14:01:20 yark (2760) yark (2760) 8 yark (2760) Definition msc 58E40 CentralizersInAlgebra