I cannot understand Artin's example of a homomorphism from $S_4$ to $S_3$ in Examples 2.5.13.
He first partitions the indices $\{1,2,3,4\}$ into three pairs of two vertices:
$$
\Pi_1 = \{1,2\} \cup \{3,4\}, \; \Pi_2 = \{1,3\} \cup \{2,4\}, \; \Pi_3 = \{1,4\}, \{2,3\}.
$$
He argues that $\sigma \in S_4$, permuting the indices, also permutes the partitions, which makes sense to me. I can take some $\sigma \in S_4$, permute the indices individually, and then observe the effect on the partitions. Because it's a bijection, I'll make a partition into a partition. So the net effect of $p = (1234)$ is $(\Pi_1, \Pi_3)$, which I believe.
My problem is: how is this a homomorphism from $S_4$ to $S_3$? Is the idea to identify $S_4$ with an isomorphic group, these pairs of partitions, and then construct a homomorphism between $S_4$ and that group? I'm struggling to see exactly what the argument is.
Best Answer
Take $\sigma\in S_4$ and consider the $6$ sets $\{1,2\},\{3,4\},\{1,3\},\{2,4\},\{1,4\}$ and $\{2,3\}$. Then, since $\sigma$ is a bijection and these are the only subsets of $\{1,2,3,4,5,6\}$ with exactly $2$ elements, $\sigma$ maps $\{1,2\}$ into exactly one of these sets and it does the same thing with $\{3,4\}$. Furthermore, since $\sigma$ is a bijection, it maps $\{1,2\}$ and $\{3,4\}$ into two disjoint sets. But then one and only of of these possibilities takes place:
So, $\sigma$ acts as a permutation of the set $\{\Pi_1,\Pi_2,\Pi_3\}$. And this defines a group homomorphism from $S_4$ into $S_3$, since $S_3$ is the group of permutations of a set with $3$ elements.