let G be abelian to find the conjugacy class of any element x we find

for any g in G

or to the definition suppose that y is conjugate to x that means there exist g in G such that

but G is abelian

if an element x is conjugate to itself that mean x in the center of G

and since G is abelian the center of G is G itself

in our case

and since G abelian all g elements are in Z(G) (i.e all elements are conjugate to itself )