I am reading through Michael Artin's Algebra, and the following passages seem contradictory to me.
On page 55: "The elements of the image correspond bijectively to the nonempty fibres, which are the equivalence classes."
On page 56: Proposition 2.7.15: "Let K be the kernel homomorphism $\phi: G \rightarrow G'$. The fibre of $\phi$ that contains an element $a$ of $G$ is the coset $aK$ of $K$. These cosets partition the group $G$, and they correspond to elements of the image of $\phi$."
I would like to know what else is in the equivalence class of a? It could be alone, but it could also be b, and b~a, so $\phi(b) = \phi(a)$. However, if the partition of $aK$ all equal $\phi(a)$, then how can $\phi$ be a bijective map? Is it because everything in $aK$ is just the same thing? If this is the case, then groups are sets, and I was under the impression that in sets, repetition of elements is not allowed.
If anyone could clear my confusion, that would be very much appreciated. If there are simple examples that demonstrate what the textbook is trying to say, that would also be very helpful.