# Thread: Equivalence Classes with a Single Element

1. ## 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.

2. 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.