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Definition (Centralizer Of An Element)
Let $a\in G$. The centralizer of $a$ is the collection of all $x\in G$ such that $ax=xa$, or alternately $x^{-1}ax=a$. We write this set symbolically as $$C(a)=\{x\in G\mid ax=xa\} = \{x\in G\mid x^{-1}ax=a\} .$$
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