I would like help with the following problem in Dummit and Foote: Let $P$ be a partition of the elements of the group $G$. we define the set consisting of the sets of the partition $P$ is a group under the following product we assume to be well-defined:
If set $S_1$ contains $s_1$ and set $S_2$ contains $s_2$ then we define $S_1S_2$ to be the unique set of $P$ containing $s_1s_2$. Prove this group is isomorphic to a quotient group of $G$ modulo $N$ where $N$ is a normal subgroup of $G$
So far I have only found trivial things: like the fact the identity must be the part containing 1. Or the fact the identity must be a subgroup of $G$. this is question 43 of section 3.1 of dummit foote.