Let $G$ be a group. If $N$ is a normal subgroup of $G$, the set of cosets of $N$ in $G$ form a group under set multiplication - this is the quotient group $G/ N$.
I'm trying to improve my intuition for the concept of a quotient group, which seems rather unnatural to me at first. While considering the property that a quotient group partitions the original group, I arrived at the following question motivated by the converse idea:
Suppose $G$ is a group. If $P$ is a partition of $G$ such that $P$ forms a group (under set multiplication), is there necessarily some normal subgroup $N \le G$ such that $P$ is the quotient group $G/N$?
This may well be trivial, but I could not see how to prove it (or find a counterexample).
My intuition is as follows: a normal subgroup of $G$ in some way characterises part of the 'structure' of $G$, and this gives rise to 'structure' in the set of its cosets. If a partition also exhibits 'structure', this element of structure will presumably be present in some subgroup of $G$, and this would suggest a relation between the two. I'm not really sure that this is meaningful, but if anyone could either correct or elaborate upon this thinking that would be very helpful.