# Equivalence Classes with a Single Element

• Mar 3rd 2011, 10:15 AM
WolfTecc
Equivalence Classes with a Single Element
Hi guys, question here. It's not for a class.

"Let $\displaystyle G$ be a group. Then the relation $\displaystyle a \sim b$ if and only if $\displaystyle a = g^{-1}bg$ for some $\displaystyle g \in G$ is an equivalence relation on $\displaystyle G$. Some equivalence classes contain only one element $\displaystyle c$. Characterize those elements $\displaystyle c$."

It's pretty clear that if $\displaystyle c$ is in the center of $\displaystyle G$, then $\displaystyle c$ must be in a class by itself since $\displaystyle b \sim c$ means $\displaystyle b = g^{-1}cg = g^{-1}g c = c$ so that $\displaystyle b = c$.

But could there be more? This question comes after a discussion of the commutator, and the book hasn't even discussed center yet, so I'm trying to figure out what kind of answer the author was after.
• Mar 3rd 2011, 11:10 AM
FernandoRevilla
Quote:

Originally Posted by WolfTecc
It's pretty clear that if $\displaystyle c$ is in the center of $\displaystyle G$, then $\displaystyle c$ must be in a class by itself

Coversely, if $\displaystyle c\not\in Z_G$ then, there exists $\displaystyle x\neq c$ such that $\displaystyle xc\neq cx$ which implies $\displaystyle c\neq x^{-1}cx$. But $\displaystyle c\equiv x^{-1}cx$ so, $\displaystyle [c]$ has at least two elements.