# Thread: Conjugacy and Abelian Groups

1. ## Conjugacy and Abelian Groups

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 $\in$ G. The element x is said to be a conjugate of the element x if there exists an a $\in$ G such that y = ${ax}a^{-1}$

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 - starting from the definition 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>

Peter

2. ## Re: Conjugacy and Abelian Groups

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

$g x g^{-1} = gg^{-1} x = x \;\;$ 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

$y = g x g^{-1}$ but G is abelian $y = gg^{-1} x \Rightarrow y =x$

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 $Z(G) = \{ g \in G \mid g.a = a \;\;\;\forall a \in G \}$
in our case $Z(G) = \{ g \in G \mid gag^{-1} = a \;\; \forall a\in G \}$
and since G abelian all g elements are in Z(G) (i.e all elements are conjugate to itself )