Chapter 3.2, Problem 11E

Let $G$ be a group.

Prove that the relation $R$ on $G$, defined by $xRy$ if and only if there exist an $a\in G$ such that $y=a^{-1}xa$, is an equivalence relation.

Let $x\in G$. Find $\left[x\right]$, the equivalence class containing $x$, if $G$ is abelian. (Sec $3.3,\#23\ll$)

Sec. 3.3, #23:

23. Let $R$ be the equivalence relation on $G$ defined by $xRy$ if and only if there exists an element $a$ in $G$ such that $y=a^{-1}xa$. If $x\in \mathbb{Z}\left(G\right)$, find $\left[x\right]$, the equivalence class containing $x$.