Why is every conjugacy class of an abelian group composed of a single element?

Question

I know that if a group is generated by a single element then the group is abelian but does this mean that if a group is abelian then its conjugacy class is composed of a single element?

Answer 1

Hint: $a$ is conjugate to $b$ if there exists some $c$ so that $b=cac^{-1}$. In an abelian group, $ac^{-1}=c^{-1}a$. Can you use associativity to conclude something interesting about the relationship between $b$ and $a$?

Answer 2

For any group $G$, the conjugacy class $[g]$ of any $g \in G$ is defined to be the set of all elements of $G$ which are conjugate to $g$, that is

$[g] = \{ xgx^{-1} \mid x \in G \}; \tag 1$

if $G$ is abelian, then

$xgx^{-1} = xx^{-1}g = e g = g \tag 2$

where $e \in G$ is the identity element; thus, by virtue of (2) we easily see that

$[g] = \{g\}, \tag 3$

the singleton set whose only element is $g$.