How to show that $C_G(a)=N_G(a)$?

Question

For singleton sets, then how show to that $C_G(a)=N_G(a)$?

As I know something about singleton set that is its closed in real line. Now I'm confused that how can I relate this concept in group theory.

I found this article on http://www.wikiwand.com/en/Centralizer_and_normalizer

Any hints/solution will be appreciated.

Thank you.

Answer 1

Let $x\in G$. We have

\begin{align} x\in C_G(a)&\iff ax=xa\\ &\iff \{a\}x=x\{a\}\quad (\text{as sets})\\ &\iff x\in N_G(a). \end{align}

Hence $C_G(a)=N_G(a)$.