Let G be a group and a an element of G. the centralizer of a is the set C(a)={g in G: ga=ag}. prove that C(a) is a subgroup of G.
Let $\displaystyle x,y\in C(a)$. Then $\displaystyle x=axa^{-1}$ and $\displaystyle y=aya^{-1}$. So $\displaystyle xy=axa^{-1}aya^{-1}=axya^{-1}$, and $\displaystyle xy\in C(a)$. Now observe that $\displaystyle (ax^{-1}a^{-1})x=(ax^{-1}a^{-1})(axa^{-1})=e$. So $\displaystyle x^{-1}=ax^{-1}a^{-1}$, that is, $\displaystyle x^{-1}\in C(a)$. Since clearly $\displaystyle e\in C(a)$, then it follows that $\displaystyle C(a)$ is a subgroup of $\displaystyle G$