I was reading Artin's Algebra and stumbled upon this example of a homomorphism $\phi:S_4 \to S_3$. See here and here for the example.
My question is what motivates the partition of the set into pairs of subsets of order 2. The text never bothers to explain the reason behind the move.
In addition, the writer claims that "An element of the symmetric group $S_4$ permutes the four indices, and by doing so it also permutes these three partitions." What does it even mean to permute the partitions? How does permuting the partitions correspond to permuting the indices?
Then I got completely lost when he says $(1\ 2\ 3\ 4)$ acts on the set $\{\Pi_1,\Pi_2,\Pi_3\}$ as $(\Pi_1\ \Pi_3)$. Exactly how are they the same?
Best Answer
To my mind it's cleaner to describe these things geometrically when possible.
$S_4$ naturally acts by rotations and reflections on a tetrahedron, permuting the four vertices. If we want to write down a map $S_4 \to S_3$ in a geometric way, we should find a set of $3$ things in a tetrahedron that rotations and reflections permute.
The clue is that a tetrahedron has $6$ edges: in fact the elements of $S_4$ naturally permute the set of pairs of opposite edges (all of which are visible in the above picture). This corresponds exactly to these partitions into pairs of subsets of size $2$, and hopefully everything else should be much clearer once you have this picture.
When Artin says permuting the indices permutes partitions, he means, for example, that if we have a permutation like $(123)$ that sends $1 \mapsto 2, 2 \mapsto 3, 3 \mapsto 1, 4 \mapsto 4$, it also acts on subsets of $\{ 1, 2, 3, 4 \}$ by permuting each individual element, so for example it sends the subset $\{ 1, 2 \}$ to the subset $\{ 2, 3 \}$, etc. and in this way it also permutes set partitions, e.g. sending the partition $12 | 34$ to the partition $23 | 14$. Again all of these can be visualized in terms of edges of the tetrahedron.