- Show that \(C_G(G) = Z(G)\).
Solution
1. By definition,\[ C_G(G)=\{g\in G:gx=xg\text{ for every }x\in G\}. \]
The centralizer \(C_G(S)\) of a subset \(S\subseteq G\) consists of the elements of \(G\) commuting with every element of \(S\). Taking \(S=G\) gives the displayed set.
2. By definition,\[ Z(G)=\{g\in G:gx=xg\text{ for every }x\in G\}. \]
This is the definition of the center of a group.
3. Therefore \(C_G(G)=Z(G)\).
The two subsets described in 1 and 2 have exactly the same membership condition.