Beachy and Blair in the book Abstract Algebra section 7.2 Conjugacy define conjugacy as follows:

Let G be a group and let x, y G. The element x is said to be a conjugate of the element x if there exists an a G such that y =

They then make the statement that "in an abelian group, elements or subgroups are only conjugate to themselves"

I have seen some brief justifications of this statement in other texts but cannot see how they link to the definition.

Can someone please -and proceeding from there - give a formal proof of this statement and then go on to show that (asserted in many texts) the conjugacy classes of an abelian group are all singleton sets>starting from the definition

Peter