# Why do the conjugate classes of a group partition the group?

Given an element a in a group G,
class(a) = {all x in G such that there exists a g in G such that gxg^(-1) = a}

class(b) = {all x in G such that there exists a g in G such that gxg^(-1) = b}

so let's say y is a conjugate of both a and b, so it is in both class(a) and class(b), does that mean that class(a) = class(b)?

given there is an element y that is in both class(a) and class(b), couldn't there be an element q that is in one class and not the other?