Center of Group and Conjugacy Classes

I am trying to prove that the center of a group G is the union of the trivial conjugacy classes of G.

So far what I have:

We know the center Z($G$) of group $G$ is defined by {$b \in G$ | $ba= ab$ for all $a \in G$}. We also know if there is an element $x \in G$ such that $b = x^{-1}ax$, then a is conjugate to b.

Since we know the elements b of G commute with every element of G in the center, we may say every element in Z($G$) has a conjugate. Therefore, the center is the union of all these conjugacy classes.

Is this correct, or where did I run off course?

• trivial conjugacy classes of G yes – Rory Dec 16 '14 at 8:08

Hint 1: If $b\in Z(G)$ and $x\in G$, what is $xbx^{-1}$?
Hint 2: In the definition of $Z(G)$, write the condition "$ba=ab$" in a way that makes one side a conjugate of $b$.
• Okay so with Hint 1, that gives me b is conjugate to b. Hint 2 gives me ba = ab -> $a^{-1}ba = b$ -> b is conjugate to b, but how does this help me say that Z(G) is the union of these conjugates. I am confused why an element being conjugate to itself helps me – Rory Dec 16 '14 at 21:44
• @Rory It shows that $b$ is in the center if and only if every conjugate of $b$ is $b$. – zibadawa timmy Dec 17 '14 at 2:39