Disclaimer: I am still new to the subject of conjugacy classes, class equations and centralizers. Some guidance to solving this problem would be greatly appreciated on my part. I would like to know how to approach the following question:
Show that a finite group has exactly one conjugacy class if and only if it is trivial.
To some, the proof may be trivial (no pun intended), but to me, it's not. I want to understand what is going on here. I would imagine that if the question were to say "Show that a finite group has exactly $2$ conjugacy classes if anf only if it has order $2$", then we would apply a similar argument. So, if this were the case, then we would see some sort of pattern, but in order to move beyond, I need to understand how to approach my initial problem.