In the book "Algebra: Chapter 0" (Paolo Aluffi), Chapter II.7 (Quotient groups), it says that given an equivalence relation $\sim$ on a group $G$, for the operation to be well-defined "in the first factor", we have: $\forall g \in G, a\sim a', \Rightarrow ag\sim a'g$, and to be well-defined "in the second factor", we have: $\forall g \in G, a\sim a', \Rightarrow ga\sim ga'$.
I am a little bit confused about these sentences. Does it mean that for every equivalence relation on a group, it has these two properties? Or it means for the equivalence relation defined on a group, we need to make sure that it has these two extra properties?
It means that these conditions have to be satisfied in order for the operation on $G/{\sim}$ (the set of equivalence classes) to be well defined.
Not every equivalence relation will satisfy these properties, so the set-theoretic quotient will not always inherit a group structure.