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

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

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 $c$ is in the center of $G$, then $c$ must be in a class by itself

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