This subset is called centralizer of $x$ and it is often denoted
$$C_G(x)=\{a\in G; xa=ax\}.$$
You can verify that this is a subgroup like this:
- $C_G(x)\ne\emptyset$ since $e\in C_G(x)$
- $a,b\in C_G(x)$ $\Rightarrow$ $(ax=xa) \land (bx=xb)$ $\Rightarrow$ $(ab)x=a(bx)=a(xb)=(ax)b=(xa)b=x(ab)$ $\Rightarrow$ $ab\in C_G(x)$
- $a\in C_G(x)$ $\Rightarrow$ $ax=xa$ $\Rightarrow$ $axa^{-1}=x$ $\Rightarrow$ $xa^{-1}=a^{-1}x$ $\Rightarrow$ $a^{-1}\in C_G(x)$
You can find a proof that it is a subgroup also at ProofWiki.
Another useful thing to notice is that $x\in C_G(x)$ and thus the subgroup generated by $x$ is in centralizer, $\langle x \rangle \subseteq C_G(x)$.