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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.

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Hint: Let $\phi\colon G \to P$ be the mapping that sends each element in $G$ to the corresponding set that it belongs to in $P$. It's not too hard to see that $\phi$ is a surjective homomorphism. Thus, it follows by the First Isomorphism Theorem that: $$ G / \ker \phi \approx \phi(G) = P $$ which is what we want, since kernels are always normal subgroups.

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You've determined that the part of $P$ containing the identity, call it $N$, is a subgroup. Next on the list of things to prove: (1) $N$ is normal, (2) parts of $P$ are cosets of $N$, (3) the group operation.

(1): You want to show $gNg^{-1}=N$ for any given $g\in G$. Consider $[g]\cdot N\cdot [g^{-1}]$.

(2) Say $g\in G,n\in N$. You want to show $[gn]=[g]\cdot N$. Which element do they both have?

(3) Finally, you want to show multiplication in $P$ corresponds to multiplication in $G/N$. Can you write out what this means? Once you have it written out, it should be straightforward to argue.

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